Chapter 6. Mixers

In this chapter, our study of building blocks focuses on downconversion and upconversion mixers, which appear in the receive path and the transmit path, respectively. While a decade ago, most mixers were realized as a Gilbert cell, many more variants have recently been introduced to satisfy the specific demands of different RX or TX architectures. In other words, a stand-alone mixer design is no longer meaningful because its ultimate performance heavily depends on the circuits surrounding it. The outline of the chapter is shown below.

6.1 General Considerations

Mixers perform frequency translation by multiplying two waveforms (and possibly their harmonics). As such, mixers have three distinctly different ports. Figure 6.1 shows a generic transceiver environment in which mixers are used. In the receive path, the down-conversion mixer senses the RF signal at its “RF port” and the local oscillator waveform at its “LO port.” The output is called the “IF port” in a heterodyne RX or the “baseband port” in a direct-conversion RX. Similarly, in the transmit path, the upconversion mixer input sensing the IF or the baseband signal is called the IF port or the baseband port, and the output port is called the RF port. The input driven by the LO is called the LO port.

Figure 6.1 Role of mixers in a generic transceiver.

How linear should each input port of a mixer be? A mixer can simply be realized as depicted in Fig. 6.2(a), where V_LO turns the switch on and off, yielding V_IF = V_RF or V_IF = 0. As explained in Chapter 2, with abrupt switching, the operation can be viewed as multiplication of the RF input by a square wave toggling between 0 and 1, even if V_LO itself is a sinusoid. Thus, as illustrated in Fig. 6.2(b), the circuit mixes the RF input with all of the LO harmonics, producing what we called “mixing spurs” in Chapter 4. In other words, the LO port of this mixer is very nonlinear. The RF port, of course, must remain sufficiently linear to satisfy the compression and/or intermodulation requirements.

Figure 6.2 (a) Mixer using an ideal switch, (b) input and output spectra.

The reader may wonder if the LO port of mixers can be linearized so as to avoid mixing with the LO harmonics. As seen later in this chapter, mixers suffer from a lower gain and higher noise as the switching in the LO port becomes less abrupt. We therefore design mixers and LO swings to ensure abrupt switching and deal with mixing spurs at the architecture level (Chapter 4).

6.1.1 Performance Parameters

Let us now consider mixer performance parameters and their role in a transceiver.

Noise and Linearity

In a receive chain, the input noise of the mixer following the LNA is divided by the LNA gain when referred to the RX input. Similarly, the IP₃ of the mixer is scaled down by the LNA gain. (Recall from Chapter 5 that the mixer noise and IP₃ are divided by different gains.) The design of downconversion mixers therefore entails a compromise between the noise figure and the IP₃ (or P_1dB). Also, the designs of the LNA and the mixer are inextricably linked, requiring that the cascade be designed as one entity.

Where in the design space do we begin then? Since the noise figure of mixers is rarely less than 8 dB, we typically allocate a gain of 10 to 15 dB to the LNA and proceed with the design of the mixer, seeking to maximize its linearity while not raising its NF. If the resulting mixer design is not satisfactory, some iteration becomes necessary. For example, we may decide to further linearize the mixer even if the NF increases and compensate for the higher noise by raising the LNA gain. We elaborate on these points in various design examples in this chapter.

In direct-conversion receivers, the IP₂ of the LNA/mixer cascade must be maximized. In Section 6.4, we introduce methods of raising the IP₂ in mixers. Also, as mentioned in Chapter 4, the mixing spurs due to the LO harmonics become important in broadband receivers.

For upconversion mixers, the noise proves somewhat critical only if the TX output noise in the RX band must be very small (Chapter 4), but even such cases demand more relaxed mixer noise performance than receivers do. The linearity of upconversion mixers is specified by the type of modulation and the baseband signal swings.

Gain

Downconversion mixers must provide sufficient gain to adequately suppress the noise contributed by subsequent stages. However, low supply voltages make it difficult to achieve a gain of more than roughly 10 dB while retaining linearity. Thus, the noise of stages following the mixer still proves critical.

In direct-conversion transmitters, it is desirable to maximize the gain and hence the output swings of upconversion mixers, thereby relaxing the gain required of the power amplifier. In two-step transmitters, on the other hand, the IF mixers must provide only a moderate gain so as to avoid compressing the RF mixer.

The gain of mixers must be carefully defined to avoid confusion. The “voltage conversion gain” of a downconversion mixer is given by the ratio of the rms voltage of the IF signal to the rms voltage of the RF signal. Note that these two signals are centered around two different frequencies. The voltage conversion gain can be measured by applying a sinusoid at ω_RF and finding the amplitude of the downconverted component at ω_IF. For upconversion mixers, the voltage conversion gain is defined in a similar fashion but from the baseband or IF port to the RF port.

In traditional RF and microwave design, mixers are characterized by a “power conversion gain,” defined as the output signal power divided by the input signal power. But in modern RF design, we prefer to employ voltage quantities because the input impedances are mostly imaginary, making the use of power quantities difficult and unnecessary.

Port-to-Port Feedthrough

Owing to device capacitances, mixers suffer from unwanted coupling (feedthrough) from one port to another [Fig. 6.3(a)]. For example, if the mixer is realized by a MOSFET [Fig. 6.3(b)], then the gate-source and gate-drain capacitances create feedthrough from the LO port to the RF and IF ports.

Figure 6.3 (a) Feedthrough mechanisms in a mixer, (b) feedthrough paths in a MOS mixer.

The effect of mixer port-to-port feedthrough on the performance depends on the architecture. Consider the direct-conversion receiver shown in Fig. 6.4. As explained in Chapter 4, the LO-RF feedthrough proves undesirable as it produces both offsets in the baseband and LO radiation from the antenna. Interestingly, this feedthrough is entirely determined by the symmetry of the mixer circuit and LO waveforms (Section 6.2.2). The LO-IF feedthrough is benign because it is heavily suppressed by the baseband low-pass filter(s).

Figure 6.4 Effect of LO-RF feedthrough.

The generation of dc offsets can also be seen intuitively. Suppose, as shown in Fig. 6.6, the RF input is a sinusoid having the same frequency as the LO. Then, each time the switch turns on, the same portion of the input waveform appears at the output, producing a certain average.

Figure 6.6 Offset generated by LO leakage.

The RF-LO and RF-IF feedthroughs also prove problematic in direct-conversion receivers. As shown in Fig. 6.7, a large in-band interferer can couple to the LO and injection-pull it (Chapter 8), thereby corrupting the LO spectrum. To avoid this effect, a buffer is typically interposed between the LO and the mixer. Also, as explained in Chapter 4, the RF-IF feedthrough corrupts the baseband signal by the beat component resulting from even-order distortion in the RF path. (This phenomenon is characterized by the IP₂.)

Figure 6.7 Effect of RF-LO feedthrough in a direct-conversion receiver.

Now, consider the heterodyne RX depicted in Fig. 6.8. Here, the LO-RF feedthrough is relatively unimportant because (1) the LO leakage falls outside the band and is attenuated by the selectivity of the LNA, the front-end band-select filter, and the antenna; and (2) the dc offset appearing at the output of the RF mixer can be removed by a high-pass filter. The LO-IF feedthrough, on the other hand, becomes serious if ω_IF and ω_LO are too close to allow filtering of the latter. The LO feedthrough may then desensitize the IF mixers if its level is comparable with their 1-dB compression point.

Figure 6.8 Effect of LO feedthrough in a heterodyne RX.

The port-to-port feedthroughs of upconversion mixers are less critical, except for the LO-RF component. As explained in Chapter 4, the LO (or carrier) feedthrough corrupts the transmitted signal constellation and must be minimized.

6.1.2 Mixer Noise Figures

The noise figure of downconversion mixers is often a source of great confusion. For simplicity, let us consider a noiseless mixer with unity gain. As shown in Fig. 6.10, the spectrum sensed by the RF port consists of a signal component and the thermal noise of R_S in both the signal band and the image band. Upon downconversion, the signal, the noise in the signal band, and the noise in the image band are translated to ω_IF. Thus, the output SNR is half the input SNR if the two noise components have equal powers, i.e., the mixer exhibits a flat frequency response at its input from the image band to the signal band. We therefore say the noise figure of a noiseless mixer is 3 dB. This quantity is called the “single-sideband” (SSB) noise figure to indicate that the desired signal resides on only one side of the LO frequency, a common case in heterodyne receivers.

Figure 6.10 SSB noise figure.

Now, consider the direct-conversion mixer shown in Fig. 6.11. In this case, only the noise in the signal band is translated to the baseband, thereby yielding equal input and output SNRs if the mixer is noiseless. The noise figure is thus equal to 0 dB. This quantity is called the “double-sideband” (DSB) noise figure to emphasize that the input signal resides on both sides of ω_LO, a common situation in direct-conversion receivers.

Figure 6.11 DSB noise figure.

In summary, the SSB noise figure of a mixer is 3 dB higher than its DSB noise figure if the signal and image bands experience equal gains at the RF port of the mixer. Typical noise figure meters measure the DSB NF and predict the SSB value by simply adding 3 dB.

NF of Direct-Conversion Receivers

It is difficult to define a noise figure for receivers that translate the signal to a zero IF (even in a heterodyne system). To understand the issue, let us consider the direct-conversion topology shown in Fig. 6.13. We recognize that the noise observed in the I output consists of the amplified noise of the LNA plus the noise of the I mixer. (The mixer DSB NF is used here because the signal spectrum appears on both sides of ω_LO.) Similarly, the noise in the Q output consists of the amplified noise of the LNA plus the noise of the Q mixer.

Figure 6.13 Direct-conversion RX for NF calculation.

But, how do we define the overall noise figure? Even though the system has two output ports, one may opt to define the NF with respect to only one,

(6.5)

where SNR_I and SNR_Q denote the SNRs measured at the I and Q outputs, respectively. Indeed, this is the most common NF definition for direct-conversion receivers. However, since the I and Q outputs are eventually combined (possibly in the digital domain), the SNR in the final combined output would serve as a more accurate measure of the noise performance. Unfortunately, the manner in which the outputs are combined depends on the modulation scheme, thus making it difficult to obtain the output SNR. For example, as described in Chapter 4, an FSK receiver may simply sample the binary levels in the I output by the data edges in the Q output, leading to a nonlinear combining of the baseband quadrature signals. For these reasons, the NF is usually obtained according to Eq. (6.5), a somewhat pessimistic value because the signal component in the other output is ignored. Ultimately, the sensitivity of the receiver is characterized by the bit error rate, thereby avoiding the NF ambiguity.

The above example leads to an important conclusion: if white noise is switched on and off with 50% duty cycle, then the resulting spectrum is still white but carries half the power. More generally, if white noise is turned on for ΔT seconds and off for T − ΔT seconds, then the resulting spectrum is still white and its power is scaled by ΔT/T. This result proves useful in the study of mixers and oscillators.

6.1.3 Single-Balanced and Double-Balanced Mixers

The simple mixer of Fig. 6.2(a) and its realization in Fig. 6.3(b) operate with a single-ended RF input and a single-ended LO. Discarding the RF signal for half of the LO period, this topology is rarely used in modern RF design. Figure 6.15(a) depicts a more efficient approach whereby two switches are driven by differential LO phases, thus “commutating” the RF input to the two outputs. Called a “single-balanced” mixer because of the balanced LO waveforms, this configuration provides twice the conversion gain of the mixer of Fig. 6.2(a) (Section 6.2.1). Furthermore, the circuit naturally provides differential outputs even with a single-ended RF input, easing the design of subsequent stages. Also, as seen in Fig. 6.15(b), the LO-RF feedthrough at ω_LO vanishes if the circuit is symmetric.¹

Figure 6.15 (a) Single-balanced passive mixer, (b) implementation of (a).

The single-balanced mixer of Fig. 6.15(b) nonetheless suffers from significant LO-IF feedthrough. In particular, denoting the coupling of V_LO to V_out1 by + αV_LO and that from to V_out2 by −αV_LO, we observe that V_out1 − V_out2 contains an LO leakage equal to 2αV_LO. To eliminate this effect, we connect two single-balanced mixers such that their output LO feedthroughs cancel but their output signals do not. Shown in Fig. 6.16, such a topology introduces two opposing feedthroughs at each output, one from V_LO and another from . The output signals remain intact because, when V_LO is high, and , and when is high, and . That is, V_out1 − V_out2 is equal to for a high LO and for a low LO.

Figure 6.16 Double-balanced passive mixer.

Called a “double-balanced” mixer, the circuit of Fig. 6.16 operates with both balanced LO waveforms and balanced RF inputs. It is possible to apply a single-ended RF input (e.g., if the LNA is single-ended) while grounding the other, but at the cost of a higher input-referred noise.

Ideal LO Waveform

What is the “ideal” LO waveform, a sinusoid or a square wave? Since each LO in an RF transceiver drives a mixer,² we note from the above observations that the LO waveform must ideally be a square wave to ensure abrupt switching and hence maximum conversion gain. For example, in the circuit of Fig. 6.16(b), if V_LO and vary gradually, then they remain approximately equal for a substantial fraction of the period (Fig. 6.17). During this time, all four transistors are on, treating V_RF as a common-mode input. That is, the input signal is “wasted” because it produces no differential component for roughly 2ΔT seconds each period. As explained later, the gradual edges may also raise the noise figure.

Figure 6.17 LO waveforms showing when the switches are on simultaneously.

At very high frequencies, the LO waveforms inevitably resemble sinusoids. We therefore choose a relatively large amplitude so as to obtain a high slew rate and ensure a minimum overlap time, ΔT.

Since mixers equivalently multiply the RF input by a square wave, they can down-convert interferers located at the LO harmonics, a serious issue in broadband receiver. For example, an interferer at 3f_LO is attenuated by about only 10 dB as it appears in the baseband.

Passive and Active Mixers

Mixers can be broadly categorized into “passive” and “active” topologies; each can be realized as a single-balanced or a double-balanced circuit. We study these types in the following sections.

6.2 Passive Downconversion Mixers

The mixers illustrated in Figs. 6.15 and 6.16 exemplify passive topologies because their transistors do not operate as amplifying devices. We wish to determine the conversion gain, noise figure, and input impedance of a certain type of passive mixers. We first assume that the LO has a duty cycle of 50% and the RF input is driven by a voltage source.

6.2.1 Gain

Let us begin with Fig. 6.18(a) and note that the input is multiplied by a square wave toggling between 0 and 1. The first harmonic of this waveform has a peak amplitude of 2/π and can be expressed as (2/π) cos ω_LOt. In the frequency domain, this harmonic consists of two impulses at ±ω_LO, each having an area of 1/π. Thus, as shown in Fig. 6.18(b), the convolution of an RF signal with these impulses creates the IF signal with a gain of 1/π (≈ −10 dB). The conversion gain is therefore equal to 1/π for abrupt LO switching. We call this topology a “return-to-zero” (RZ) mixer because the output falls to zero when the switch turns off.

Figure 6.18 (a) Input and output waveforms of a return-to-zero mixer, (b) corresponding spectra.

The reader may wonder why resistor R_L is used in the circuit of Fig. 6.18(a). What happens if the resistor is replaced with a capacitor, e.g., the input capacitance of the next stage? Depicted in Fig. 6.21(a) and called a “sampling” mixer or a “non-return-to-zero” (NRZ) mixer, such an arrangement operates as a sample-and-hold circuit and exhibits a higher gain because the output is held—rather than reset—when the switch turns off. In fact, the output waveform of Fig. 6.21(a) can be decomposed into two as shown in Fig. 6.21(b), where y₁(t) is identical to the return-to-zero output in Fig. 6.18(a), and y₂(t) denotes the additional output stored on the capacitor when S₁ is off. We wish to compute the voltage conversion gain.

Figure 6.21 (a) Sampling mixer, (b) output waveform decomposition.

We first recall the following Fourier transform pairs:

(6.9)-(6.11)

where Π[t/(T/2) − 1/2] represents a square pulse with an amplitude of 1 between t = 0 and t = T/2 and zero elsewhere. The right-hand side of Eq. (6.11) can also be expressed as a sinc. Since y₁(t) is equal to x(t) multiplied by a square wave toggling between zero and 1, and since such a square wave is equal to the convolution of a square pulse and a train of impulses [Fig. 6.22(a)], we have

(6.12)

Figure 6.22 (a) Decomposition of a square wave, (b) input and output spectra corresponding to y₁(t).

where T_LO denotes the LO period. It follows from Eqs. (6.9) and (6.11) that

(6.13)

Figure 6.22(b) shows the corresponding spectra. The component of interest in Y₁(f) lies at the IF and is obtained by setting k to ±1:

(6.14)

The impulse, in essence, computes [1/(jω)][1 − exp(−jωT_LO/2)] at ±1/T_LO, which amounts to ±T_LO/(jπ). Multiplying this result by (1/T_LO)δ(f ± 1/T_LO) and convolving it with X(f), we have

(6.15)

As expected, the conversion gain from X(f) to Y₁(f) is equal to 1/π, but with a phase shift of 90°.

The second output in Fig. 6.21(b), y₂(t), can be viewed as a train of impulses that sample the input and are subsequently convolved with a square pulse [Fig. 6.23(a)]. That is,

(6.16)

Figure 6.23 (a) Decomposition of y₂(t), (b) corresponding spectrum.

and hence

(6.17)

Figure 6.23(b) depicts the spectrum, revealing that shifted replicas of X(f) are multiplied by a sinc envelope. Note the subtle difference between Y₁(f) and Y₂(f): in the former, each replica of X(f) is simply scaled by a factor, whereas in the latter, each replica experiences a “droop” due to the sinc envelope. The component of interest in Y₂(f) is obtained by setting k to ±1:

(6.18)

The term in the second set of square brackets must be calculated at the IF. If the IF is much lower than 2f_LO, then exp(−jω_IFT_LO/2) ≈ 1 − jω_IFT_LO/2. Thus,

(6.19)

Note that Y₂(f) in fact contains a larger IF component than does Y₁(f). The total IF output is therefore equal to

(6.20)-(6.21)

If realized as a single-balanced topology (Fig. 6.24), the circuit provides a gain twice this value, 1.186 ≈ 1.48 dB. That is, a single-balanced sampling mixer exhibits about 5.5 dB higher gain than its return-to-zero counterpart. It is remarkable that, though a passive circuit, the single-ended sampling mixer actually has a voltage conversion gain greater than unity, and hence is a more attractive choice. The return-to-zero mixer is rarely used in modern RF design.

Figure 6.24 Single-balanced sampling mixer.

The above example may rule out the use of double-balanced sampling mixers. Since most receiver designs incorporate a single-ended LNA, this is not a serious limitation. However, if necessary, double-balanced operation can be realized through the use of two single-balanced mixers whose outputs are summed in the current domain. Illustrated conceptually in Fig. 6.26 [1], the idea is to retain the samples on the capacitors, convert each differential output voltage to a current by means of M₁–M₄, add their output currents, and apply the currents to load resistors, thus generating an output voltage. In this case, the mixer conversion gain is still equal to 1.48 dB.

Figure 6.26 Output combining of two single-balanced mixers in the current domain.

6.2.2 LO Self-Mixing

Recall from Chapter 4 that the leakage of the LO waveform to the input of a mixer is added to the RF signal and mixed with the LO, generating a dc offset at the output. We now study this mechanism in the single-balanced sampling mixer. Consider the arrangement shown in Fig. 6.27(a), where R_S denotes the output impedance of the previous stage (the LNA). Suppose the LO waveforms and the transistors are perfectly symmetric. Then, due to the nonlinearity of C_GS1 and C_GS2 arising from large LO amplitudes, V_P does change with time but only at twice the LO frequency [Fig. 6.27(b)] (Problem 6.3). Upon mixing with the LO signal, this component is translated to f_LO and 3f_LO—but not to dc. In other words, with perfectly-symmetric devices and LO waveforms, the mixer exhibits no LO self-mixing and hence no output dc offsets.

Figure 6.27 (a) LO-RF leakage path in a sampling mixer, (b) LO and leakage waveforms.

In practice, however, mismatches between M₁ and M₂ and within the oscillator circuit give rise to a finite LO leakage to node P. Accurate calculation of the resulting dc offset is difficult owing to the lack of data on various transistor, capacitor, and inductor mismatches that lead to asymmetries. A rough rule of thumb is 10–20 millivolts at the output of the mixer.

6.2.3 Noise

In this section, we study the noise behavior of return-to-zero and sampling mixers. Our approach is to determine the output noise spectrum, compute the output noise power in 1 Hz at the IF, and divide the result by the square of the conversion gain, thus obtaining the input-referred noise.

Let us begin with the RZ mixer, shown in Fig. 6.28. Here, R_on denotes the on-resistance of the switch. We assume a 50% duty cycle for the LO. The output noise is given by 4kT(R_on||R_L) when S₁ is on and by 4kTR_L when it is off. As shown in Example 6.4, on the average, the output contains half of 4kT(R_on||R_L) and half of 4kTR_L:

(6.22)

Figure 6.28 RZ mixer for noise calculation.

If we select R_on R_L so as to minimize the conversion loss, then

(6.23)

Dividing this result by 1/π², we have

(6.24)-(6.25)

That is, the noise power of R_L (= 4kTR_L) is “amplified” by a factor of 5 when referred to the input.

The reader may wonder if our choice R_on R_L is optimum. If R_L is very high, the output noise decreases but so does the conversion gain. We now remove the assumption R_on R_L and express the voltage conversion gain as (1/π)R_L/(R_on + R_L). Dividing Eq. (6.22) by the square of this value gives

(6.27)

This function reaches a minimum of

(6.28)-(6.29)

for . For example, if R_on = 100 Ω and , then the input-referred noise voltage is equal to (equivalent to an NF of 17.7 dB in a 50-Ω system).

In reality, the output noise voltages calculated above are pessimistic because the input capacitance of the following stage limits the noise bandwidth, i.e., the noise is no longer white. This point becomes clearer in our study of the sampling mixer.

We now wish to compute the output noise spectrum of a sampling mixer. The output noise at the IF can then be divided by the conversion gain to obtain the input-referred noise voltage. We begin with three observations. First, in the simple circuit of Fig. 6.29(a) (where R₁ denotes the switch resistance), if V_in = 0,

(6.30)

Figure 6.29 (a) Equivalent circuit of sampling mixer for noise calculations, (b) noise in on and off states, and (c) decomposition of output waveform.

where (for − ∞ < ω < + ∞). We say the noise is “shaped” by the filter.³ Second, in the switching circuit of Fig. 6.29(b), the output is equal to the shaped noise of R₁ when S₁ is on and a sampled, constant value when it is off. Third, in a manner similar to the gain calculation in Fig. 6.21, we can decompose the output into two waveforms V_n1 and V_n2 as shown in Fig. 6.29(c).

It is tempting to consider the overall output spectrum as the sum of the spectra of V_n1 and V_n2. However, as explained below, the low-frequency noise components generated by R₁ create correlation between the track-mode and hold-mode noise waveforms. For this reason, we proceed as follows: (1) compute the spectrum of V_n1 while excluding the low-frequency components in the noise of R₁, (2) do the same for V_n2, and (3) add the contribution of the low-frequency components to the final result. In the derivations below, we refer to the first two as simply the spectra of V_n1 and V_n2 even though V_n1(t) and V_n2(t) in Fig. 6.29 are affected by the low-frequency noise of R₁. Similarly, we use the notation even though its low-frequency components are removed and considered separately.

Spectrum of V_n1

To calculate the spectrum of V_n1, we view this waveform as the product of V_n,LPF(t) and a square wave toggling between 0 and 1. As shown in Fig. 6.30, the spectrum of V_n1 is given by the convolution of and the power spectral density of the square wave (impulses with a sinc² envelope). In practice, the sampling bandwidth of the mixer, 1/(R₁C₁), rarely exceeds 3ω_LO, and hence

(6.31)

where the factor of 2 on the right-hand side accounts for the aliasing of components at negative and positive frequencies. At low output frequencies, this expression reduces to

(6.32)

Note that this is the two-sided spectrum of .

Figure 6.30 Aliasing in V_n1.

Spectrum of V_n2

The spectrum of V_n2 in Fig. 6.29(c) can be obtained using the approach illustrated in Fig. 6.21 for the conversion gain. That is, V_n2 is equivalent to sampling V_n,LPF by a train of impulses and convolving the result with a square pulse, Π[t/(2T_LO) − 1/2]. We must therefore convolve the spectrum of V_n,LPF with a train of impulses (each having an area of ) and multiply the result by a sinc² envelope. As shown in Fig. 6.31, the convolution translates noise components around ±f_LO, ±2f_LO, etc., to the IF. The sum of these aliased components is given by

(6.33)-(6.34)

where a = 2πR₁C₁f_LO. For the summation in Eq. (6.34), we have

(6.35)

Also, typically (2πR₁C₁)⁻¹ > f_LO and hence coth(2R₁C₁f_LO)⁻¹ ≈ 1. It follows that

(6.36)

Figure 6.31 Aliasing in V_n2.

This result must be multiplied by the sinc² envelope, |(jω)⁻¹[1 − exp(−jωT_LO/2)]|², which has a magnitude of at low frequencies. Thus, the two-sided IF spectrum of V_n2 is given by

(6.37)

Correlation Between V_n1 and V_n2

We must now consider the correlation between V_n1 and V_n2 in Fig. 6.29. The correlation arises from two mechanisms: (1) as the circuit enters the track mode, the previous sampled value takes a finite time to vanish, and (2) when the circuit enters the hold mode, the frozen noise value, V_n2, is partially correlated with V_n1. The former mechanism is typically negligible because of the short track time constant. For the latter, we recognize that the noise frequency components far below f_LO remain relatively constant during the track and hold modes (Fig. 6.32); it is as if they experienced a zero-order hold operation and hence a conversion gain of unity. Thus, the R₁ noise components from 0 to roughly f_LO/10 directly appear at the output, adding a noise PSD of 2kTR₁.

Figure 6.32 Correlation between noise components in acquisition and hold modes.

Summing the one-sided spectra of V_n1 and V_n2 and the low-frequency contribution, 4kTR₁, gives the total (one-sided) output noise at the IF:

(6.38)

The input-referred noise is obtained by dividing this result by 1/π² + 1/4:

(6.39)

Note that [2] and [3] do not predict the dependence on R₁ or C₁.

For a single-balanced topology, the differential output exhibits a noise power twice that given by Eq. (6.38), but the voltage conversion gain is twice as high. Thus, the input-referred noise of a single-balanced passive (sampling) mixer is equal to

(6.40)-(6.41)

Let us now study the noise of a double-balanced passive mixer. As mentioned in Example 6.8, the behavior of the circuit does not depend much on the absence or presence of load capacitors. With abrupt LO edges, a resistance equal to R₁ appears between one input and one output at any point in time [Fig. 6.33(a)]. Thus, from Fig. 6.33(b), . Since the voltage conversion is equal to 2/π,

(6.42)

Figure 6.33 (a) Equivalent circuit of double-balanced passive mixer, (b) simplified circuit.

The low gain of passive mixers makes the noise of the subsequent stage critical. Figure 6.34(a) shows a typical arrangement, where a quasi-differential pair (Chapter 5) serves as an amplifier and its input capacitance holds the output of the mixer. Each common-source stage exhibits an input-referred noise voltage of

(6.43)

Figure 6.34 (a) Passive mixer followed by gain stage, (b) bias path at the RF input, (c) bias path at the baseband output.

This power should be doubled to account for the two halves of the circuit and added to the mixer output noise power.

How is the circuit of Fig. 6.34(a) biased? Depicted in Fig. 6.34(b) is an example. Here, the bias of the preceding stage (the LNA) is blocked by C₁, and the network consisting of R_REF, M_REF, and I_REF defines the bias current of M₁ and M₂. As explained in Chapter 5, resistor R_REF is chosen much greater than the output resistance of the preceding stage. We typically select W_REF ≈ 0.2W_1,2 so that I_D1,2 ≈ 5I_REF.

In the circuit of Fig. 6.34(b), the dc voltages at nodes A and B are equal to V_P unless LO self-mixing produces a dc offset between these two nodes. The reader may wonder if the circuit can be rearranged as shown in Fig. 6.34(c) so that the bias resistors provide a path to remove the dc offset. The following example elaborates on this point.

6.2.4 Input Impedance

Passive mixers tend to present an appreciable load to LNAs. We therefore wish to formulate the input impedance of passive sampling mixers.

Consider the circuit depicted in Fig. 6.36, where S₁ is assumed ideal for now. Recall from Fig. 6.21 that the output voltage can be viewed as the sum of two waveforms y₁(t) and y₂(t), given by Eqs. (6.12) and (6.16), respectively. The current drawn by C₁ in Fig. 6.36 is equal to

(6.44)

Figure 6.36 Input impedance of sampling mixer.

Moreover, i_in(t) = i_out(t). Taking the Fourier transform, we thus have

(6.45)

where Y(f) is equal to the sum of Y₁(f) and Y₂(f).

As evident from Figs. 6.22 and 6.23, Y(f) contains many frequency components. We must therefore reflect on the meaning of the “input impedance.” Since the input voltage signal, x(t), is typically confined to a narrow bandwidth, we seek frequency components in I_in(f) that lie within the bandwidth of x(t). To this end, we set k in Eqs. (6.13) and (6.17) to zero so that X(f) is simply convolved with δ(f) [i.e., the center frequency of X(f) does not change]. (This stands in contrast to gain and noise calculations, where k was chosen to translate X(f) to the IF of interest.) It follows that

(6.46)

In the square brackets in the first term, ω must be set to zero to evaluate the impulse at f = 0. Thus, the first term reduces to (1/2)X(f). In the second term, the exponential in the square brackets must also be calculated at ω = 0. Consequently, the second term simplifies to (1/T_LO)X(f)[1/(jω)][1−exp(−jωT_LO/2)]. We then arrive at an expression for the input admittance:

(6.47)

Note that the on-resistance of the switch simply appears in series with the inverse of (6.47).

It is instructive to examine Eq. (6.47) for a few special cases. If ω (the input frequency) is much less than ω_LO, then the second term in the square brackets reduces to 1/2 and

(6.48)

In other words, the entire capacitance is seen at the input [Fig. 6.37(a)]. If ω ≈ 2πf_LO (as in direct-conversion receivers), then the second term is equal to 1/(jπ) and

(6.49)

Figure 6.37 Input impedance of passive mixer for (a) ω ω_LO and (b) ω ≈ ω_LO.

The input impedance thus contains a parallel resistive component equal to 1/(2fC₁) [Fig. 6.37(b)]. Finally, if ω 2πf_LO, the second term is much less than the first, yielding

(6.50)

For the input impedance of a single-balanced mixer, we must add the switch on-resistance, R₁, to the inverse of Eq. (6.47) and halve the result. If ω ≈ ω_LO, then

(6.51)

Flicker Noise

An important advantage of passive mixers over their active counterparts is their much lower output flicker noise. This property proves critical in narrowband applications, where 1/f noise in the baseband can substantially corrupt the downconverted channel.

MOSFETs produce little flicker noise if they carry a small current [4], a condition satisfied in a passive sampling mixer if the load capacitance is relatively small. However, the low gain of passive mixers makes the 1/f noise contribution of the subsequent stage critical. Thus, the baseband amplifier following the mixer must employ large transistors, presenting a large load capacitance to the mixer (Fig. 6.38). As explained above, C_BB manifests itself in the input impedance of the mixer, Z_mix, thereby loading the LNA.

Figure 6.38 Baseband input capacitance reflected at the input of passive mixer.

LO Swing

Passive MOS mixers require large (rail-to-rail) LO swings, a disadvantage with respect to active mixers. Since LC oscillators typically generate large swings, this is not a serious drawback, at least at moderate frequencies (up to 5 or 10 GHz).

In Chapter 13, we present the design of a passive mixer followed by a baseband amplifier for 11a/g applications.

6.2.5 Current-Driven Passive Mixers

The gain, noise, and input impedance analyses carried out in the previous sections have assumed that the RF input of passive mixers is driven by a voltage source. If driven by a current source, such mixers exhibit different properties. Figure 6.39(a) shows a conceptual arrangement where the LNA has a relatively high output impedance, approximating a current source. The passive mixer still carries no bias current so as to achieve low flicker noise and it drives a general impedance Z_BB. Voltage-driven and current-driven passive mixers entail a number of interesting differences.

Figure 6.39 (a) Current-driven passive mixer, (b) simplified model for input impedance calculation, (c) spectra at input and output.

First, the input impedance of the current-driven mixer in Fig. 6.39 is quite different from that of the voltage-driven counterpart. The reader may find this strange. Indeed, familiar circuits exhibit an input impedance that is independent of the source impedance: we can calculate the input impedance of an LNA by applying a voltage or a current source to the input port. A passive mixer, on the other hand, does not satisfy this intuition because it is a time-variant circuit. To determine the input impedance of a current-driven single-balanced mixer, we consider the simplified case depicted in Fig. 6.39(b), where the on-resistance of the switches is neglected. We wish to calculate Z_in(f) = V_RF(f)/I_in(f) in the vicinity of the carrier (LO) frequency, assuming a 50% duty cycle for the LO.

The input current is routed to the upper arm for 50% of the time and flows through Z_BB. In the time domain [5],

(6.52)

where S(t) denotes a square wave toggling between 0 and 1, and h(t) is the impulse response of Z_BB. In the frequency domain,

(6.53)

where S(f) is the spectrum of a square wave. As expected, upon convolution with the first harmonic of S(f), I_in(f) is translated to the baseband and is then subjected to the frequency response of Z_BB(f). A similar phenomenon occurs in the lower arm.

We now make a critical observation [5]: the switches in Fig. 6.39(b) also mix the baseband waveforms with the LO, delivering the upconverted voltages to node A. Thus, V₁(t) is multiplied by S(t) as it returns to the input, and its spectrum is translated to RF. The spectrum of V₂(t) is also upconverted and added to this result.

Figure 6.39(c) summarizes our findings, revealing that the downconverted spectrum of I_in(f) is shaped by the frequency response of Z_BB, and the result “goes back” through the mixer, landing around f_c while retaining its spectral shape. In other words, in response to the spectrum shown for I_in(f), an RF voltage spectrum has appeared at the input that is shaped by the baseband impedance. This implies that the input impedance around f_c resembles a frequency-translated version of Z_BB(f). For example, if Z_BB(f) is a low-pass impedance, then Z_in(f) has a band-pass behavior [5].

The second property of current-driven passive mixers is that their noise and nonlinearity contribution are reduced [6]. This is because, ideally, a device in series with a current source does not alter the current passing through it.

Passive mixers need not employ a 50% LO duty cycle. In fact, both voltage-driven and current-driven mixers utilizing a 25% duty cycle provide a higher gain. Figure 6.40 shows the quadrature LO waveforms according to this scenario. Writing the Fourier series for LO waveforms having a duty cycle of d, the reader can show that the RF current entering each switch generates an IF current given by [6]:

(6.54)

where I_RF0 denotes the peak amplitude of the RF current. As expected, d = 0.5 yields a gain of 2/π. More importantly, for d = 0.25, the gain reaches , 3 dB higher. Of course, the generation of these waveforms becomes difficult at very high frequencies. [Ideally, we would choose d ≈ 0 (impulse sampling) to raise this gain to unity.]

Figure 6.40 Quadrature LO waveforms with 25% duty cycle.

Another useful attribute of the 25% duty cycle in Fig. 6.40 is that the mixer switches driven by LO₀ and LO₁₈₀ (or by LO₉₀ and LO₂₇₀) are not on simultaneously. As a result, the mixer contributes smaller noise and nonlinearity [6].

6.3 Active Downconversion Mixers

Mixers can be realized so as to achieve conversion gain in one stage. Called active mixers, such topologies perform three functions: they convert the RF voltage to a current, “com-mutate” (steer) the RF current by the LO, and convert the IF current to voltage. These operations are illustrated in Fig. 6.41. While both passive and active mixers incorporate switching for frequency translation, the latter precede and follow the switching by voltage-to-current (V/I) and current-to-voltage (I/V) conversion, respectively, thereby achieving gain. We can intuitively observe that the input transconductance, I_RF/V_RF, and the output transresistance, V_IF/I_IF, can, in principle, assume arbitrarily large values, yielding an arbitrarily high gain.

Figure 6.41 Active mixer viewed as a V/I converter, a current switch, and an I/V converter.

Figure 6.42 depicts a typical single-balanced realization. Here, M₁ converts the input RF voltage to a current (and is hence called a “transconductor”), the differential pair M₂–M₃ commutates (steers) this current to the left and to the right, and R₁ and R₂ convert the output currents to voltage. We call M₂ and M₃ the “switching pair.” As with our passive mixer study in Section 6.2, we wish to quantify the gain, noise, and nonlinearity of this circuit. Note that the switching pair does not need rail-to-rail LO swings. In fact, as explained later, such swings degrade the linearity.

Figure 6.42 Single-balanced active mixer.

Double-Balanced Topology

If the RF input is available in differential form, e.g., if the LNA provides differential outputs, then the active mixer of Fig. 6.42 must be modified accordingly. We begin by duplicating the circuit as shown in Fig. 6.43(a), where and denote the differential phases of the RF input. Each half circuit commutates the RF current to its IF outputs. Since , the small-signal IF components at X₁ and Y₁ are equal to the negative of those at X₂ and Y₂, respectively. That is, V_X1 = −V_Y1 = −V_X2 = V_Y2, allowing us to short X₁ to Y₂ and X₂ to Y₁ and arrive at the double-balanced mixer in Fig. 6.43(b), where the load resistors are equal to R_D/2. We often draw the circuit as shown in Fig. 6.43(c) for the sake of compactness. Transistors M₂, M₃, M₅, and M₆ are called the “switching quad.” We will study the advantages and disadvantages of this topology in subsequent sections.

Figure 6.43 (a) Two single-balanced mixers sensing differential RF inputs, (b) summation of output currents, (c) compact drawing of circuit.

One advantage of double-balanced mixers over their single-balanced counterparts stems from their rejection of amplitude noise in the LO waveform. We return to this property in Section 6.3.2.

6.3.1 Conversion Gain

In the circuit of Fig. 6.42, transistor M₁ produces a small-signal drain current equal to g_m1 V_RF. With abrupt LO switching, the circuit reduces to that shown in Fig. 6.44(a), where M₂ multiplies I_RF by a square wave toggling between 0 and 1, S(t), and M₃ multiplies I_RF by S(t − T_LO/2) because LO and are complementary. It follows that

(6.55)

(6.56)

Figure 6.44 (a) Equivalent circuit of active mixer, (b) switching waveforms.

Since V_out = V_DD − I₁R₁ − (V_DD − I₂R₂), we have for R₁ = R₂ = R_D,

(6.57)

From Fig. 6.44(b), we recognize that the switching operation in Eq. (6.57) is equivalent to multiplying I_RF by a square wave toggling between −1 and + 1. Such a waveform exhibits a fundamental amplitude equal to 4/π,⁴ yielding an output given by

(6.58)

If I_RF(t) = g_m1V_RF cos ω_RFt, then the IF component at ω_RF − ω_LO is equal to

(6.59)

The voltage conversion gain is therefore equal to

(6.60)

What limits the conversion gain? We assume a given power budget, i.e., a certain bias current, I_D1, and show that the gain trades with the linearity and voltage headroom. The input transistor is sized according to the overdrive voltage, V_GS1 − V_TH1, that yields the required IP₃ (Chapter 5). Thus, V_DS1,min = V_GS1 − V_TH1. The transconductance of M₁ is limited by the current budget and IP₃, as expressed by g_m1 = 2I_D1/(V_GS1 − V_TH1) [or I_D1/(V_GS1 − V_TH1) for velocity-saturated devices]. Also, the value of R_D is limited by the maximum allowable dc voltage across it. In other words, we must compute the minimum allowable value of V_X and V_Y in Fig. 6.42. As explained in Section 6.3.3, linearity requirements dictate that M₂ and M₃ not enter the triode region so long as both carry current.

Suppose the gate voltages of M₂ and M₃ in Fig. 6.42 are held at the common-mode level of the differential LO waveforms, V_CM,LO [Fig. 6.45(a)]. If M₁ is at the edge of saturation, then V_N ≥ V_GS1 − V_TH1:

(6.61)

Figure 6.45 (a) Active mixer with LO at CM level, (b) required swing to turn one device off.

Now consider the time instant at which the gate voltages of M₂ and M₃ reach V_CM,LO + V₀ and V_CM,LO − V₀, respectively, where , a value high enough to turn off M₃ [Fig. 6.45(b)]. For M₂ to remain in saturation up to this point, its drain voltage must not fall below :

(6.62)

which, from Eq. (6.61), reduces to

(6.63)

Thus, V_X,min must accommodate the overdrive of M₁ and about 1.7 times the “equilibrium” overdrive of each of the switching transistors. The maximum allowable dc voltage across each load resistor is equal to

(6.64)

Since each resistor carries half of I_D1,

(6.65)

From (6.64) and (6.65), we obtain the maximum voltage conversion gain as

(6.66)-(6.67)

We therefore conclude that low supply voltages severely limit the gain of active mixers.

How much room for improvement do we have? Given by IP₃ requirements, the over-drive of the input transistor has little flexibility unless the gain of the preceding LNA can be reduced. This is possible if the mixer noise figure can also be lowered, which, as explained in Section 6.3.2, trades with the power dissipation and input capacitance of the mixer. The equilibrium overdrive of the switching transistors can be reduced by making the two transistors wider (while raising the capacitance seen at the LO port).

The conversion gain may also fall if the LO swing is lowered. As illustrated in Fig. 6.46, while M₂ and M₃ are near equilibrium, the RF current produced by M₁ is split approximately equally between them, thus appearing as a common-mode current and yielding little conversion gain for that period of time. Reduction of the LO swing tends to increase this time and lower the gain (unless the LO is a square wave).

Figure 6.46 RF current as a CM component near LO zero crossings.

With a sinusoidal LO, the drain currents of the switching devices depart from square waves, remaining approximately equal for a fraction of each half cycle, ΔT [Fig. 6.48(a)]. As mentioned previously, the circuit exhibits little conversion gain during these periods. We now wish to estimate the reduction in the gain.

Figure 6.48 (a) Effect of gradual LO transitions, (b) magnified LO waveforms.

A differential pair having an equilibrium overdrive of (V_GS − V_TH)_eq steers most of its tail current for a differential input voltage, ΔV_in, of (for square-law devices). We assume that the drain currents are roughly equal for ΔV_in ≤ (V_GS − V_TH)_eq/5 and calculate the corresponding value of ΔT. We note from Fig. 6.48(b) that, if each single-ended LO waveform has a peak amplitude of V_p,LO, then LO and reach a difference of (V_GS − V_TH)_eq/5 in approximately ΔT/2 = (V_GS − V_TH)_eq/5/(2V_p,LOω_LO) seconds. Multiplying this result by a factor of 4 to account for the total time on both rising and falling edges and normalizing to the LO period, we surmise that the overall gain of the mixer is reduced to

(6.71)-(6.72)

The second phenomenon that degrades the gain relates to the total capacitance seen at the drain of the input transistor. Consider an active mixer in one-half of the LO cycle (Fig. 6.49). With abrupt LO edges, M₂ is on and M₃ is off, yielding a total capacitance at node P equal to

(6.75)

Figure 6.49 Loss of RF current to ground through C_P.

Note that C_GS3 is substantially smaller than C_GS2 in this phase (why?). The RF current produced by M₁ is split between C_P and the resistance seen at the source of M₂, 1/g_m2 (if body effect is neglected). Thus, the voltage conversion gain is reduced by a factor of g_m2/(sC_P + g_m2); i.e., Eq. (6.72) must be modified as

(6.76)

How significant is this current division? In other words, how does compare with in the above expression? Note that g_m2/C_P is well below the maximum f_T of M₂ because (a) the sum of C_DB1, C_SB2, C_SB3, and C_GS3 is comparable with or larger than C_GS2, and (b) the low overdrive voltage of M₂ (imposed by headroom and gain requirements) also leads to a low f_T. We therefore observe that the effect of C_P may become critical for frequencies higher than roughly one-tenth of the maximum f_T of the transistors.

6.3.2 Noise in Active Mixers

The analysis of noise in active mixers is somewhat different from the study undertaken in Section 6.2.3 for passive mixers. As illustrated conceptually in Fig. 6.51, the noise components of interest lie in the RF range before downconversion and in the IF range after downconversion. Note that the frequency translation of RF noise by the switching devices prohibits the direct use of small-signal ac and noise analysis in circuit simulators (as is done for LNAs), necessitating simulations in the time domain. Moreover, the noise contributed by the switching devices exhibits time-varying statistics, complicating the analysis.

Figure 6.51 Partitioning of active mixer for noise analysis.

Qualitative Analysis

To gain insight into the noise behavior of active mixers, we begin with a qualitative study. Let us first assume abrupt LO transitions and consider the representation in Fig. 6.52(a) for half of the LO cycle. Here,

(6.78)

Figure 6.52 (a) Effect of noise when one transistor is off, (b) equivalent circuit of (a).

In this phase, the circuit reduces to a cascode structure, with M₂ contributing some noise because of the capacitance at node P (Chapter 5). Recall from the analysis of cascode LNAs in Chapter 5 that, at frequencies well below f_T, the output noise current generated by M₂ is equal to V_n,M2C_Ps [Fig. 6.52(b)]. This noise and the noise current of M₁ (which is dominant) are multiplied by a square wave toggling between 0 and 1. Transistor M₃ plays an identical role in the next half cycle of the LO.

Now consider a more realistic case where the LO transitions are not abrupt, allowing M₂ and M₃ to remain on simultaneously for part of the period. As depicted in Fig. 6.53, the circuit now resembles a differential pair near equilibrium, amplifying the noise of M₂ and M₃—while the noise of M₁ has little effect on the output because it behaves as a common-mode disturbance.

Figure 6.53 Effect of noise of M₂ and M₃ near equilibrium.

It is important to make an observation regarding the mixer of Fig. 6.53. The noise generated by the local oscillator and its buffer becomes indistinguishable from the noise of M₂ and M₃ when these two transistors are around equilibrium. As depicted in Fig. 6.55, a differential pair serving as the LO buffer may produce an output noise much higher than that of M₂ and M₃. It is therefore necessary to simulate the noise behavior of mixers with the LO circuitry present.

Figure 6.55 Effect of LO buffer noise on single-balanced mixer.

Quantitative Analysis

Consider the single-balanced mixer depicted in Fig. 6.51. From our qualitative analysis, we identify three sections in the circuit: the RF section, the time-varying (switching) section, and the IF section. To estimate the input-referred noise voltage, we apply the following procedure: (1) for each source of noise, determine a “conversion gain” to the IF output; (2) multiply the magnitude of each noise by the corresponding gain and add up all of the resulting powers, thus obtaining the total noise at the IF output; (3) divide the output noise by the overall conversion gain of the mixer to refer it to the input.

Let us begin the analysis by assuming abrupt LO transitions with a 50% duty cycle. In each half cycle of the LO, the circuit resembles that in Fig. 6.57, i.e., the noise of M₁ (I_n1,M1) and each of the switching devices is multiplied by a square wave toggling between 0 and 1. We have seen in Example 6.4 that, if white noise is switched on and off with 50% duty cycle, the resulting spectrum is still white while carrying half of the power. Thus, half of the noise powers (squared current quantities) of M₁ and M₂ is injected into node X, generating an output noise spectral density given by , where denotes the noise current injected by M₂ into node X. The total noise at node X is therefore equal to

(6.82)

Figure 6.57 Noise of input device and one switching device in an active mixer.

The noise power must be doubled to account for that at node Y and then divided by the square of the conversion gain. From Eq. (6.76), the conversion gain in the presence of a capacitance at node P is equal to for abrupt LO edges (i.e., if V_p,LO → ∞). Note that the C_P’s used for the noise contribution of M₂ and gain calculation are given by (6.75) and (6.78), respectively, and slightly different. Nonetheless, we assume they are approximately equal. The input-referred noise voltage is therefore given by

(6.83)-(6.84)

If the effect of C_P is negligible, then

(6.85)

The term π²kTγ/g_m1 in (6.85) represents the input-referred contribution of M₁. This appears puzzling: why is this contribution simply not equal to the gate-referred noise of M₁,4kTγ/g_m1? We investigate this point in Problem 6.7.

We now consider the effect of gradual LO transitions on the noise behavior. Similar to the gain calculations in Section 6.3.1, we employ a piecewise-linear approximation (Fig. 6.58): the switching transistors are considered near equilibrium for 2 ΔT = 2(V_GS − V_TH)_eq/(5V_P,LOω_LO) seconds per LO cycle, injecting noise to the output as a differential pair. During this time period, M₁ contributes mostly common-mode noise, and the output noise is equal to

(6.87)

where we assume g_m2 ≈ g_m3. Now, this noise power must be weighted by a factor of 2ΔT/T_LO, and that in the numerator of Eq. (6.83) by a factor of 1 − 2ΔT/T_LO. The sum of these weighted noise powers must then be divided by the square of (6.76) to refer it to the input. The input-referred noise is thus given by

(6.88)

Figure 6.58 Piecewise-linear waveforms for mixer noise calculation.

Equation (6.88) reveals that the equilibrium overdrive voltage of the switching devices plays a complex role here: (1) in the first term in the numerator, (for a given bias current), whereas ΔT ∝ (V_GS − V_TH)_eq; (2) the noise power expressed by the second term in the numerator is proportional to 1 − 2 ΔT/T_LO while the squared gain in the denominator varies in proportion to (1 − 2 ΔT/T_LO)², suggesting that ΔT must be minimized.

Flicker Noise

Unlike passive mixers, active topologies suffer from substantial flicker noise at their output, a serious issue if the IF signal resides near zero frequency and has a narrow bandwidth.

Consider the circuit shown in Fig. 6.60(a). With perfect symmetry, the 1/f noise of I_SS does not appear at the output because it is mixed with ω_LO (and its harmonics). Thus, only the flicker noise of M₂ and M₃ must be considered. The noise of M₂, , experiences the gain of the differential pair as it propagates to the output. Fortunately, the large LO swing heavily saturates (desensitizes) the differential pair most of the time, thereby lowering the gain seen by .

Figure 6.60 (a) Flicker noise of switching device, (b) LO and drain current waveforms, (c) modulation of zero crossing due to flicker noise, (d) equivalent pulsewidth modulation.

In order to compute the gain experienced by V_n2 in Fig. 6.60(a), we assume a sinusoidal LO but also a small switching time for M₂ and M₃ such that I_SS is steered almost instantaneously from one to the other at the zero crossings of LO and [Fig. 6.60(b)]. How does V_n2 alter this behavior? Upon addition to the LO waveform, the noise modulates the zero crossings of the LO [7]. This can be seen by computing the time at which the gate voltages of M₁ and M₂ are equal; i.e., by equating the instantaneous gate voltages of M₂ and M₃:

(6.89)

obtaining

(6.90)

In the vicinity of t = 0, we have

(6.91)

The crossing of LO and is displaced from its ideal point by an amount of ΔT (ω_LO ΔT 1 rad) [Fig. 6.60(c)]:

(6.92)

That is,

(6.93)

Note that 2V_p,LOω_LO is the slope of the differential LO waveform,⁵ S_LO, and hence |ΔT| = |V_n2(t)|/S_LO.

We now assume nearly abrupt drain current switching for M₂ and M₃ and consider the above zero-crossing deviation as pulsewidth modulation of the currents [Fig. 6.60(d)]. Drawing the differential output current as in Fig. 6.60(d), we note that the modulated output is equal to the ideal output plus a noise waveform consisting of a series of narrow pulses of height 2I_SS and width ΔT and occurring twice per period [7]. If each narrow pulse is approximated by an impulse, the noise waveform in I_D2 − I_D3 can be expressed as

(6.94)

In the frequency domain, from Eq. (6.9),

(6.95)

The baseband component is obtained for k = 0 because V_n2(f) has a low-pass spectrum. It follows that

(6.96)

and hence

(6.97)

In other words, the flicker noise of each transistor is scaled by a factor of I_SSR_D/(πV_p,LO) as it appears at the output. It is therefore desirable to minimize the bias current of the switching devices. Note that this quantity must be multiplied by to account for the flicker noise of M₃ as well.

The above study also explains the low 1/f noise of passive mixers. Since I_SS = 0 in passive topologies, a noise voltage source in series with the gate experiences a high attenuation as it appears at the output. (Additionally, MOSFETs carrying negligible current produce negligible flicker noise.)

The reader may wonder if the above results apply to the thermal noise of M₂ and M₃ as well. Indeed, the analysis is identical [7] and the same results are obtained, with V_n2(f) replaced with 4kTγ/g_m2. The reader can show that this method and our earlier method of thermal noise analysis yield roughly equal results if πV_p,LO ≈ 5(V_GS − V_TH)_eq2,3.

Another flicker noise mechanism in active mixers arises from the finite capacitance at node P in Fig. 6.60(a) [7]. It can be shown that the differential output current in this case includes a flicker noise component given by [7]

(6.100)

Thus, a higher tail capacitance or LO frequency intensifies this effect. Nonetheless, the first mechanism tends to dominate at low and moderate frequencies.

6.3.3 Linearity

The linearity of active mixers is determined primarily by the input transistor’s overdrive voltage. As explained in Chapter 5, the IP₃ of a common-source transistor rises with the overdrive, eventually reaching a relatively constant value.

The input transistor imposes a direct trade-off between nonlinearity and noise because

(6.101)-(6.102)

We also noted in Section 6.3.1 that the headroom consumed by the input transistor, V_GS − V_TH, lowers the conversion gain [Eq. (6.67)]. Along with the above example, these observations point to trade-offs among noise, nonlinearity, gain, and power dissipation in active mixers.

The linearity of active mixers degrades if the switching transistors enter the triode region. To understand this phenomenon, consider the circuit shown in Fig. 6.61, where M₂ is in the triode region while M₃ is still on and in saturation. Note that (1) the load resistors and capacitors establish an output bandwidth commensurate with the IF signal, and (2) the IF signal is uncorrelated with the LO waveform. If both M₂ and M₃ operate in saturation, then the division of I_RF between the two transistors is given by their transconductances and is independent of their drain voltages.⁶ On the other hand, if M₂ is in the triode region, then I_D2 is a function of the IF voltage at node X, leading to signal-dependent current division between M₂ and M₃. To avoid this nonlinearity, M₂ must not enter the triode region so long as M₃ is on and vice versa. Thus, the LO swings cannot be arbitrarily large.

Figure 6.61 Effect of output waveform on current steering when one device enters the triode region.

Compression

Let us now study gain compression in active mixers. The above effect may manifest itself as the circuit approaches compression. If the output swings become excessively large, the circuit begins to compress at the output rather than at the input, by which we mean the switching devices introduce nonlinearity and hence compression while the input transistor has not reached compression. This phenomenon tends to occur if the gain of the active mixer is relatively high.

The input transistor may introduce compression even if it satisfies the quadratic characteristics of long-channel MOSFETs. This is because, with a large input level, the gate voltage of the device rises while the drain voltage falls, possibly driving it into the triode region. From Fig. 6.51, we can write the RF voltage swing at node P as

(6.103)

where R_P denotes the “average resistance” seen at the common source node of M₂ and M₃.⁷ We can approximate R_P as (1/g_m2)||(1/g_m3), where g_m2 and g_m3 represent the equilibrium transconductances of M₂ and M₃, respectively. In a typical design, g_m1R_P is on the order of unity. Thus, in the above example, as the input rises by 355 mV/2 = 178 mV from its bias value, the drain voltage of the input device falls by about 178 mV. If M₁ must not enter the triode region, then the drain-source headroom allocated to M₁ must be 355 mV higher than its quiescent overdrive voltage. Note that we did not account for this extra drain voltage swing in Example 6.12. If we had, the conversion gain would have been even lower.

The IP₂ of active mixers is also of great interest. We compute the IP₂ in Section 6.4.3.

6.4 Improved Mixer Topologies

The mixer performance envelope defined by noise, nonlinearity, gain, power dissipation, and voltage headroom must often be pushed beyond the typical scenarios studied thus far in this chapter. For this reason, a multitude of circuit techniques have been introduced to improve the performance of mixers, especially active topologies. In this section, we present some of these techniques.

6.4.1 Active Mixers with Current-Source Helpers

The principal difficulty in the design of active mixers stems from the conflicting requirements between the input transistor current (which must be high enough to meet noise and linearity specifications) and the load resistor current (which must be low enough to allow large resistors and hence a high gain). We therefore surmise that adding current sources (“helpers”) in parallel with the load resistors (Fig. 6.66) alleviates this conflict by affording larger resistor values. If I_D1 = 2I₀ and each current source carries a fraction, αI₀, then R_D can be as large as V₀/[(1 − α)I₀], where V₀ is the maximum allowable drop across R_D [as formulated by Eq. (6.64)]. Consequently, the voltage conversion gain rises as α increases. For example, if α = 0.5, then R_D can be doubled and so can the gain. A higher R_D also reduces its input-referred noise contribution [Eq. (6.85)].

Figure 6.66 Addition of load current sources to relax headroom constraints.

But how about the noise contributed by M₄ and M₅? Assuming that these devices are biased at the edge of saturation, i.e., |V_GS − V_TH|_4,5 = V₀, we write the noise current of each as 4kTγg_m = 4kTγ(2αI₀)/V₀, multiply it by to obtain the (squared) noise voltage at each output node,⁸ and sum the result with the noise of R_D itself:

(6.110)

where the noise due to other parts of the mixer is excluded. Since the voltage conversion gain is proportional to R_D, the above noise power must be normalized to [and eventually the other factors in Eq. (6.85)]. We thus write

(6.111)-(6.113)

Interestingly, the total noise due to each current-source helper and its corresponding load resistor rises with α, beginning from 4kTI₀/V₀ for α = 0 and reaching (4kTI₀/V₀)(2γ) for α = 1.

The addition of the helpers in Fig. 6.66 also degrades the linearity. In the calculations leading to Eq. (6.113), we assumed that the helpers operate at the edge of saturation so as to minimize their transconductance and hence their noise current, but this bias condition readily drives them into the triode region in the presence of signals. The circuit is therefore likely to compress at the output rather than at the input.

6.4.2 Active Mixers with Enhanced Transconductance

Following the foregoing thought process, we can insert the current-source helper in the RF path rather than in the IF path. Depicted in Fig. 6.67 [8], the idea is to provide most of the bias current of M₁ by M₄, thereby reducing the current flowing through the load resistors (and the switching transistors). For example, if |I_D4| = 0.75I_D1, then R_D and hence the gain can be quadrupled. Moreover, the reduction of the bias current switched by M₂ and M₃ translates to a lower overdrive voltage and more abrupt switching, decreasing ΔT in Figs. 6.48(a) and 6.58 and lessening the gain and noise effects formulated by Eqs. (6.72) and (6.88). Finally, the output flicker noise falls (Problem 6.10).

Figure 6.67 Addition of current source to tail of switching pair.

The above approach nonetheless faces two issues. First, transistor M₄ contributes additional capacitance to node P, exacerbating the difficulties mentioned earlier. As a smaller bias current is allocated to M₂ and M₃, raising the impedance seen at their source [≈ 1/(2g_m)], C_P “steals” a greater fraction of the RF current generated by M₁, reducing the gain. Second, the noise current of M₄ directly adds to the RF signal. We can readily express the noise currents of M₁ and M₄ as

(6.115)-(6.116)

In order to suppress the capacitance and noise contribution of M₄ in Fig. 6.68, an inductor can be placed in series with its drain. Illustrated in Fig. 6.69(a) [9], such an arrangement not only enhances the input transconductance but allows the inductor to resonate with C_P. Additionally, capacitor C₁ acts as a short at RF, shunting the noise current of M₄ to ground. As a result, most of the RF current produced by M₁ is commutated by M₂ and M₃, and the noise injected by M₂ and M₃ is also reduced (because they switch more abruptly).

Figure 6.69 (a) Use of inductive resonance at tail with helper current source, (b) equivalent circuit of inductor.

In the circuit of Fig. 6.69(a), the inductor parasitics must be managed carefully. First, L₁ contributes some capacitance to node P, equivalently raising C_P. Second, the loss of L₁ translates to a parallel resistance, “wasting” the RF current and adding noise. Depicted in Fig. 6.69(b), this resistance, R₁, must remain much greater than 1/(2g_m2,3) so as to negligibly shunt the RF current. Also, its noise current must be much less than that of M₁. Thus, the choice of the inductor is governed by the following conditions:

(6.117)-(6.119)

where C_P,tot includes the capacitance of L₁.

The circuits of Figs. 6.67 and 6.69 suffer from a drawback in deep-submicron technologies: since M₁ is typically a small transistor, it poorly matches the current mirror arrangement that feeds M₄. As a result, the exact current flowing through the switching pair may vary considerably.

Figure 6.70 shows another topology wherein capacitive coupling permits independent bias currents for the input transistor and the switching pair [10]. Here, C₁ acts as a short circuit at RF and L₁ resonates with the parasitics at nodes P and N. Furthermore, the voltage headroom available to M₁ is no longer constrained by (V_GS − V_TH)_2,3 and the drop across the load resistors. In a typical design, I_D1/I₀ may fall in the range of 3 to 5 for optimum performance. Note that if I₀ is excessively low, the switching pair does not absorb all of the RF current. Another important attribute is that, as formulated by Eq. (6.97), a smaller I₀ leads to lower flicker noise at the output.

Figure 6.70 Active mixer using capacitive coupling with resonance.

6.4.3 Active Mixers with High IP₂

As explained in Chapter 4, the second intercept point becomes critical in direct-conversion and low-IF receivers as it signifies the corruption introduced by the beating of two interferers or envelope demodulation of one interferer. We also noted that capacitive coupling between the LNA and the mixer removes the low-frequency beat, making the mixer the bottleneck. Thus, a great deal of effort has been expended on high-IP₂ mixers.

It is instructive to compute the IP₂ of a single-balanced mixer in the presence of asymmetries. (Recall from Chapter 4 that a symmetric mixer has an infinite IP₂.) Let us begin with the circuit of Fig. 6.71(a), where V_OS denotes the offset voltage associated with M₂ and M₃. We wish to compute the fraction of I_SS that flows to the output without frequency translation. As with the flicker noise calculations in Section 6.3.2, we assume LO and exhibit a finite slope but M₂ and M₃ switch instantaneously, i.e., they switch the tail current according to the sign of V_A − V_B.

Figure 6.71 (a) Active mixer with offset voltage, (b) effect of offset on LO waveforms, (c) duty cycle distortion of drain currents, (d) circuit for IP₂ computation.

As shown in Fig. 6.71(b), the vertical shift of V_LO displaces the consecutive crossings of LO and by ± ΔT, where ΔT = V_OS/S_LO and S_LO denotes the differential slope of the LO (= 2V_p,LOω_LO). This forces M₂ to remain on for T_LO/2 + 2 ΔT seconds and M₃ for T_LO/2 − 2 ΔT seconds. It follows from Fig. 6.71(c) that the differential output current, I_D2 − I_D3 contains a dc component equal to (4 ΔT/T_LO)I_SS = V_OSI_SS/(πV_p,LO), and the differential output voltage a dc component equal to V_OSI_SSR_D/(πV_p,LO). As expected, this result agrees with Eq. (6.97) because the offset can be considered a very slow noise component.

An interesting observation offered by the output 1/f noise and offset equations is as follows. If the bias current of the switching pair is reduced but that of the input transconductor is not, then the performance improves because the gain does not change but the output 1/f noise and offset fall. For example, the current helpers described in the previous section prove useful here.

We now replace I_SS with a transconductor device as depicted in Fig. 6.71(d) and assume

(6.120)

where V_GS0 is the bias gate-source voltage of M₁. With a square-law device, the IM₂ product emerges in the current of M₁ as

(6.121)

Multiplying this quantity by V_OSR_D/(πV_p,LO) yields the direct feedthrough to the output:

(6.122)

To calculate the IP₂, the value of V_m must be raised until the amplitude of V_IM2,out becomes equal to the amplitude of the main downconverted components. This amplitude is simply given by (2/π)g_m1R_DV_m. Thus,

(6.123)

Writing g_m1 as μ_nC_ox(W/L)(V_GS − V_TH)₁, we finally obtain

(6.124)

For example, if (V_GS − V_TH)₁ = 250 mV, V_p,LO = 300 mV, and V_OS = 10 mV, then V_IIP2 = 30 V_P (= 39.5 dBm in a 50-Ω system). Other IP₂ mechanisms are described in [12].

The foregoing analysis also applies to asymmetries in the LO waveforms that would arise from mismatches within the LO circuitry and its buffer. If the duty cycle is denoted by (T_LO/2 − ΔT)/T_LO (e.g., 48%), then the dc component in I_D1 − I_D2 is equal to (2 ΔT/T_LO)I_SS, yielding an average of (2 ΔT/T_LO)I_SSR_D at the output. We therefore replace I_SS with the IM₂ component given by Eq. (6.121), arriving at

(6.125)

Equating the amplitude of this component to (2/π)g_m1R_DV_m and substituting μ_nC_ox(W/L)(V_GS − V_TH)₁ for g_m1, we have

(6.126)

For example, a duty cycle of 48% along with (V_GS − V_TH)₁ = 250 mV gives rise to V_IIP2 = 7.96 V_P (= 28 dBm in a 50-Ω system).

In order to raise the IP₂, the input transconductor of an active mixer can be realized in differential form, leading to a double-balanced topology. Shown in Fig. 6.72, such a circuit produces a finite IM₂ product only as a result of mismatches between M₁ and M₂. We quantify this effect in the following example. Note that, unlike the previous double-balanced mixers, this circuit employs a tail current source.

Figure 6.72 Input offset in a double-balanced mixer.

While improving the IP₂ significantly, the use of a differential pair in Fig. 6.72 degrades the IP₃. As formulated in Chapter 5, a quasi-differential pair (with the sources held at ac ground) exhibits a higher IP₃. We now repeat the calculations leading to Eq. (6.131) for such a mixer (Fig. 6.73), noting that the input pair now has poor common-mode rejection. Let us apply and , obtaining

(6.132)

(6.133)

Figure 6.73 Effect of offsets in a double-balanced mixer using a quasi-differential input pair.

While independent of V_OS1, the low-frequency beat in I_D1 is multiplied by a factor of V_OS2R_D/(πV_p,LO) and that in I_D2 by V_OS3R_D/(πV_p,LO). Here, V_OS2 and V_OS3 denote the offsets of M₃–M₄ and M₅–M₆, respectively. The output thus exhibits an IM₂ component given by

(6.134)

Noting that the output amplitude of each fundamental is equal to (2/π)2V_mg_m1R_D and that g_m1 = μ_nC_ox(W/L)₁(V_GS0 − V_TH), we have

(6.135)

For example, if V_GS − V_TH = 250 mV, V_p,LO = 300 mV, and V_OS2 = V_OS3 = + 10 mV, then V_IIP2 = 30 V_p (= + 39.5 dBm in a 50-Ω system). Comparison of the IIP₂’s obtained for the differential and quasi-differential mixers indicates that the latter is much inferior, revealing a trade-off between IP₂ and IP₃.

We have thus far considered one mechanism leading to a finite IP₂: the passage of the low-frequency beat through the mixer’s switching devices. On the other hand, even with no even-order distortion in the transconductor, it is still possible to observe a finite low-frequency beat at the output if (a) the switching devices (or the LO waveforms) exhibit asymmetry and (b) a finite capacitance appears at the common source node of the switching devices [11, 12]. In this case, two interferers, V_m cos ω₁t + V_m cos ω₂t, arriving at the common source node experience nonlinearity and mixing with the LO harmonics, thereby generating a component at ω₁ − ω₂ after downconversion. The details of this mechanism are described in [11, 12].

While conceived for noise and gain optimization reasons, the mixer topology in Fig. 6.70 also exhibits a high IP₂. The high-pass filter consisting of L₁, C₁, and the resistance seen at node P suppresses low-frequency beats generated by the even-order distortion in M₁. From the equivalent circuit shown in Fig. 6.74, we have

(6.136)-(6.137)

Figure 6.74 Effect of low-frequency beat in a mixer using capacitive coupling and resonance.

At low frequencies, this result can be approximated as

(6.138)

revealing a high attenuation.

Another approach to raising the IP₂ is to degenerate the transconductor capacitively. As illustrated in Fig. 6.75 [10], the degeneration capacitor, C_d, acts as a short circuit at RF but nearly an open circuit at the low-frequency beat components. Expressing the transconductance of the input stage as

(6.139)-(6.140)

we recognize that the gain at low frequencies falls in proportion to C_ds, making M₁ incapable of generating second-order intermodulation components.

Figure 6.75 Effect of capacitive degeneration on IP₂.

As mentioned earlier, even with capacitive coupling between the transconductor stage and the switching devices, the capacitance at the common source node of the switching pair ultimately limits the IP₂ (if the offset of the switching pair is considered). We therefore expect a higher IP₂ if an inductor resonates with this capacitance. Figure 6.77 shows a double-balanced mixer employing both capacitive degeneration and resonance to achieve an IP₂ of + 78 dBm [11].

Figure 6.77 Use of inductor at sources of switching quad to raise IP₂.

6.4.4 Active Mixers with Low Flicker Noise

Our study of noise in Section 6.3.2 revealed that the downconverted flicker noise of the switching devices is proportional to their bias current and the parasitic capacitance at their common source node. Since these trends also hold for the IP₂ of active mixers, we postulate that the techniques described in Section 6.4.3 for raising the IP₂ lower flicker noise as well. In particular, the circuit topologies in Figs. 6.69 and 6.74 both allow a lower bias current for the switching pair and cancel the tail capacitance by the inductor. This approach, however, demands two inductors (one for each quadrature mixer), complicating the layout and routing.

Let us return to the helper idea shown in Fig. 6.67 and ask, is it possible to turn on the helper only at the time when it is needed? In other words, can we turn on the PMOS current source only at the zero crossings of the LO so that it lowers the bias current of the switching devices and hence the effect of their flicker noise [13]? In such a scheme, the helper itself would inject only common-mode noise because it turns on only when the switching pair is in equilibrium.

Figure 6.78 depicts our first attempt in realizing this concept. Since large LO swings produce a reasonable voltage swing at node P at 2ω_LO, the diode-connected transistor turns on when LO and cross and V_P falls. As LO or rises, so does V_P, turning M_H off. Thus, M_H can provide most of the bias current of M₁ near the crossing points of LO and while injecting minimal noise for the rest of the period.

Figure 6.78 Use of a diode-connected device to reduce switching pair current.

Unfortunately, the diode-connected transistor in Fig. 6.78 does not turn off abruptly as LO and depart from their crossing point. Consequently, M_H continues to present a low impedance at node P, shunting the RF current to ac ground. This issue can be resolved in a double-balanced mixer by reconfiguring the diode-connected devices as a cross-coupled pair [13]. As illustrated in Fig. 6.79 [13], M_H1 and M_H2 turn on and off simultaneously because V_P and V_Q vary identically—as if M_H1 and M_H2 were diode-connected devices. Thus, these two transistors provide most of the bias currents of M₁ and M₄ at the crossing points of LO and . On the other hand, as far as the differential RF current of M₁ and M₄ is concerned, the cross-coupled pair acts as a negative resistance (Chapter 8), partially cancelling the positive resistance presented by the switching pairs at P and Q. Thus, M_H1 and M_H2 do not shunt the RF current.

Figure 6.79 Use of cross-coupled pair to reduce current of switching quad.

The circuit of Fig. 6.79 nonetheless requires large LO swings to ensure that V_P and V_Q rise rapidly and sufficiently so as to turn off M_H1 and M_H2.¹⁰ Otherwise, these two devices continue to inject differential noise for part of the period. Another drawback of this technique is that it does not lend itself to single-balanced mixers.

The notion of reducing the current through the switching devices at the crossing points of LO and can alternatively be realized by turning off the transconductor momentarily [14]. Consider the circuit shown in Fig. 6.81(a), where switch S₁ is driven by a waveform having a frequency of 2f_LO but a duty cycle of, say, 80%. As depicted in Fig. 6.81(b), S₁ briefly turns the transconductor off twice per LO period. Thus, if the crossing points of LO and are chosen to fall at the times when I_P is zero, then the flicker noise of M₂ and M₃ is heavily attenuated. Moreover, M₂ and M₃ inject no thermal noise to the output near the equilibrium. The concept can be extended to quadrature double-balanced mixers [14]. In Problem 6.12, we decide whether this circuit can also be viewed as a differential pair whose current is modulated (chopped) at a rate of 2f_LO.

Figure 6.81 (a) Use of a switch to turn off the switching pair near LO zero crossings, (b) circuit waveforms.

The above approach entails a number of issues. First, the turn-off time of the transconductor must be sufficiently long and properly phased with respect to LO and so that it encloses the LO transitions. Second, at high frequencies it becomes difficult to generate 2f_LO with such narrow pulses; the conversion gain thus suffers because the transconductor remains off for a greater portion of the period. Third, switch S₁ in Fig. 6.81 does consume some voltage headroom if its capacitances must be negligible.

6.5 Upconversion Mixers

The transmitter architectures studied in Chapter 4 employ upconversion mixers to translate the baseband spectrum to the carrier frequency in one or two steps. In this section, we deal with the design of such mixers.

6.5.1 Performance Requirements

Consider the generic transmitter shown in Fig. 6.82. The design of the TX circuitry typically begins with the PA and moves backward; the PA is designed to deliver the specified power to the antenna while satisfying certain linearity requirements (in terms of the adjacent-channel power or 1-dB compression point). The PA therefore presents a certain input capacitance and, owing to its moderate gain, demands a certain input swing. Thus, the upconversion mixers must (1) translate the baseband spectrum to a high output frequency (unlike downconversion mixers) while providing sufficient gain, (2) drive the input capacitance of the PA, (3) deliver the necessary swing to the PA input, and (4) not limit the linearity of the TX. In addition, as studied in Chapter 4, dc offsets in upconversion mixers translate to carrier feedthrough and must be minimized.

Figure 6.82 Generic transmitter.

The interface between the mixers and the PA entails another critical issue. Since the baseband and mixer circuits are typically realized in differential form, and since the antenna is typically single-ended, the designer must decide at what point and how the differential output of the mixers must be converted to a single-ended signal. As explained in Chapter 5, this operation presents many difficulties.

The noise requirement of upconversion mixers is generally much more relaxed than that of downconversion mixers. As studied in Problem 6.13, this is true even in GSM, wherein the amplified noise of the upconversion mixers in the receive band must meet certain specifications (Chapter 4).

The interface between the baseband DACs and the upconversion mixers in Fig. 6.82 also imposes another constraint on the design. Recall from Chapter 4 that high-pass filtering of the baseband signal introduces intersymbol interference. Thus, the DACs must be directly coupled to the mixers to avoid a notch in the signal spectrum.¹¹ As seen below, this issue dictates that the bias conditions in the upconversion mixers be relatively independent of the output common-mode level of the DACs.

6.5.2 Upconversion Mixer Topologies

Passive Mixers

The superior linearity of passive mixers makes them attractive for upconversion as well. We wish to construct a quadrature upconverter using passive mixers.

Our study of downconversion mixers has revealed that single-balanced sampling topologies provide a conversion gain that is about 5.5 dB higher than their return-to-zero counterparts. Is this true for upconversion, too? Consider a low-frequency baseband sinusoid applied to a sampling mixer (Fig. 6.83). The output appears to contain mostly the input waveform and little high-frequency energy. To quantify our intuition, we return to the constituent waveforms, y₁(t) and y₂(t), given by Eqs. (6.12) and (6.16), respectively, and reexamine them for upconversion, assuming that x(t) is a baseband signal. The component of interest in Y₁(f) still occurs at k = ± 1 and is given by

(6.142)

Figure 6.83 Sampling mixer for upconversion.

For Y₂(f), we must also set k to ±1:

(6.143)

However, the term in the second set of brackets must be evaluated at the upconverted frequency. If ω = ω_LO + ω_BB, where ω_BB denotes the baseband frequency, then exp(−jωT_LO/2) = exp(−jπ) exp(−jω_BBT_LO/2), which, for ω_BB 2f_LO, reduces to −(1− jω_BBT_LO/2). Similarly, if ω = − ω_LO − ω_BB, then exp(−jωT_LO/2) ≈ − (1 + jω_BBT_LO/2). Adding Y₁(f) and Y₂(f) gives

(6.144)

indicating that the upconverted output amplitude is proportional to ω_BB/(ω_LO + ω_BB) ≈ ω_BB/ω_LO. Thus, such a mixer is not suited to upconversion.

In Problem 6.14, we study a return-to-zero mixer for upconversion and show that its conversion gain is still equal to 2/π (for a single-balanced topology). Similarly, from Example 6.8, a double-balanced passive mixer exhibits a gain of 2/π. Depicted in Fig. 6.84(a), such a topology is more relevant to TX design than single-balanced structures because the baseband waveforms are typically available in differential form. We thus focus on double-balanced mixers here.

Figure 6.84 (a) Double-balanced upconversion passive mixer, (b) use of resonance to increase bandwidth.

While simple and quite linear, the circuit of Fig. 6.84(a) must deal with a number of issues. First, the bandwidth at nodes X and Y must accommodate the upconverted signal frequency so as to avoid additional loss. This bandwidth is determined by the on-resistance of the switches (R_on), their capacitance contributions to the output nodes, and the input capacitance of the next stage (C_in). Wider switches increase the bandwidth up to the point where their capacitances overwhelm C_in, but they also present a greater capacitance at the LO ports.

It is possible to null the capacitance at nodes X and Y by means of resonance. As illustrated in Fig. 6.84(b) [15], inductor L₁ resonates with the total capacitance at X and Y, and its value is chosen to yield

(6.145)

where C_X,Y denotes the capacitances contributed by the switches at X or Y. At resonance, the mixers are loaded by the parallel equivalent resistance of the inductor, R₁ = QL₁ω_IF. Thus, we require that 2R_on R₁ to avoid additional loss. This technique becomes necessary only at very high frequencies, e.g., at 50 GHz and above.

The second issue relates to the use of passive mixers in a quadrature upconverter, where the outputs of two mixers must be summed. Unfortunately, passive mixers sense and produce voltages, making direct summation difficult. We therefore convert each output to current, sum the currents, and convert the result to voltage. Figure 6.85(a) depicts such an arrangement. Here, the quasi-differential pairs M₁–M₂ and M₃–M₄ perform V/I conversion, and the load resistors, I/V conversion. This circuit can provide gain while lending itself to low supply voltages. The grounded sources of M₁–M₄ also yield a relatively high linearity.¹²

Figure 6.85 (a) Summation of quadrature outputs, (b) bias definition issue.

A drawback of the above topology is that its bias point is sensitive to the input common-mode level, i.e., the output CM level of the preceding DAC. As shown in Fig. 6.85(b), I_D1 depends on V_BB and varies significantly with process and temperature. For this reason, we employ ac coupling between the mixer and the V/I converter and define the latter’s bias by a current mirror. Alternatively, we can resort to true differential pairs, with their common-source nodes at ac ground (Fig. 6.86). Defined by the tail currents, the bias conditions now remain relatively independent of the input CM level, but each tail current source consumes voltage headroom.

Figure 6.86 Addition of tail current to define bias of upconversion V/I converters.

The third issue concerns the available overdrive voltage of the mixer switches, a particularly serious problem in Fig. 6.85(b). We note that M₅ can be ac coupled to M₁, but still requiring a gate voltage of V_TH5 + V_GS1 + V_BB to turn on. Thus, if the peak LO level is equal to V_DD, the switch experiences an overdrive of only V_DD − (V_TH5 + V_BB), thereby suffering from a tight trade-off between its on-resistance and capacitance. A small over-drive also degrades the linearity of the switch. For example, if V_DD = 1 V, V_TH5 = 0.3 V, and V_BB = 0.5 V, then the overdrive is equal to 0.2 V. It is important to recognize that the use of inductors in Fig. 6.87 relaxes the headroom consumption from V_DD through R_D and M₁, but the headroom limitation in the path consisting of V_DD, V_GS5, and V_BB still persists.

The foregoing difficulty can be alleviated if the peak LO level can exceed V_DD. This is accomplished if the LO buffer contains a load inductor tied to V_DD (Fig. 6.88).

Figure 6.88 Mixer headroom considerations.

Now, the dc level of the LO is approximately equal to V_DD, with the peak reaching V_DD + V₀. For example, if V_DD = 1 V, V_TH5 = 0.3 V, V_BB = 0.5 V, and V₀ = 0.5 V, then the overdrive of M₅ is raised to 0.7 V.

The above-V_DD swings in Fig. 6.88 do raise concern with respect to device voltage stress and reliability. In particular, if the baseband signal has a peak amplitude of V_a and a CM level of V_BB, then the gate-source voltage of M₅ reaches a maximum of V_DD + V₀ − (V_BB−V_a), possibly exceeding the value allowed by the technology. In the above numerical example, since the overdrive of M₅ approaches 0.7 V, V_GS5 = 0.7 V + V_TH5 = 1 V in the absence of the baseband signal. Thus, if the maximum allowable V_GS is 1.2 V, the baseband peak swing is limited to 0.2 V. As explained in Chapter 4, small baseband swings exacerbate the problem of carrier feedthrough in transmitters.

It is important to note that, by now, we have added quite a few inductors to the circuit: one in Fig. 6.84(b) to improve the bandwidth, one in Fig. 6.87 to save voltage headroom, and another in Fig. 6.88 to raise the overdrive of the switches. A quadrature upconverter therefore requires a large number of inductors. The LO buffer in Fig. 6.88 can be omitted if the LO signal is capacitively coupled to the gate of M₅ and biased at V_DD.

Carrier Feedthrough

It is instructive to study the sources of carrier feedthrough in a transmitter using passive mixers. Consider the baseband interface shown in Fig. 6.89, where the DAC output contains a peak signal swing of V_a and an offset voltage of V_OS,DAC.

Figure 6.89 Effect of baseband offset in upconversion mixing.

An ideal double-balanced passive mixer upconverts both the signal and the offset, producing at its output the RF (or IF) signal and a carrier (LO) component. If modeled as a multiplier, the mixer generates an output given by

(6.146)

where α is related to the conversion gain. Expanding the right-hand side yields

(6.147)

Since α/2 = 2/π for a double-balanced mixer, we note that the carrier feedthrough has a peak amplitude of αV_OS,DAC = (4/π)V_OS,DAC. Alternatively, we recognize that the relative carrier feedthrough is equal to αV_OS,DAC/(αV_a/2) = 2V_OS,DAC/V_a. For example, if V_OS,DAC = 10mV and V_a = 0.1 V, then the feedthrough is equal to −34 dB.

Let us now consider the effect of threshold mismatches within the switches themselves. As illustrated in Fig. 6.90(a), the threshold mismatch in one pair shifts the LO waveform vertically, distorting the duty cycle. That is, is multiplied by the equivalent waveforms shown in Fig. 6.90(b). Does this operation generate an output component at f_LO? No, carrier feedthrough can occur only if a dc component in the baseband is mixed with the fundamental LO frequency. We therefore conclude that threshold mismatches within passive mixers introduce no carrier feedthrough.¹³

Figure 6.90 (a) Offset in a passive upconversion mixer, (b) effect on LO waveforms.

The carrier feedthrough in passive upconversion mixers arises primarily from mismatches between the gate-drain capacitances of the switches. As shown in Fig. 6.91, the LO feedthrough observed at X is equal to

(6.148)

where C_X denotes the total capacitance seen from X to ground (including the input capacitance of the following stage).

Figure 6.91 LO feedthrough paths in a passive mixer.

Active Mixers

Upconversion in a transmitter can be performed by means of active mixers, facing issues different from those of passive mixers. We begin with a double-balanced topology employing a quasi-differential pair (Fig. 6.92). The inductive loads serve two purposes, namely, they relax voltage headroom issues and raise the conversion gain (and hence the output swings) by nulling the capacitance at the output node. As with active downconversion mixers studied in Section 6.3, the voltage conversion gain can be expressed as

(6.149)

where R_p is the equivalent parallel resistance of each inductor at resonance.

Figure 6.92 Active upconversion mixer.

With only low frequencies present at the gates and drains of M₁ and M₂ in Fig. 6.92, the circuit is quite tolerant of capacitance at nodes P and Q, a point of contrast to downconversion mixers. However, stacking of the transistors limits the voltage headroom. Recall from downconversion mixer calculations in Section 6.3 that the minimum allowable voltage at X (or Y) is given by

(6.150)

if the dc drop across the inductors is neglected. For example, if V_GS1 − V_TH1 = 300mV and V_GS3 − V_TH3 = 200mV, then V_X,min = 640mV, allowing a peak swing of V_DD − V_X,min = 360mV at X if V_DD = 1 V. This value is reasonable.

Unfortunately, the bias conditions of the circuit of Fig. 6.92 heavily depend on the DAC output common-mode level. Thus, we apply the modification shown in Fig. 6.86, arriving at the topology in Fig. 6.94(a) (a Gilbert cell). This circuit faces two difficulties. First, the current source consumes additional voltage headroom. Second, since node A cannot be held at ac ground by a capacitor at low baseband frequencies, the nonlinearity is more pronounced. We therefore fold the input path and degenerate the differential pair to alleviate these issues [Fig. 6.94(b)].

Figure 6.94 (a) Gilbert cell as upconversion mixer, (b) mixer with folded input stage.

Despite degeneration, the circuit of Fig. 6.94(b) may experience substantial nonlinearity if the baseband voltage swing exceeds a certain value. We recognize that, if V_in1 − V_in2 becomes sufficiently negative, |I_D1| approaches I₃, starving M₃ and M₅. Now, if the differential input becomes more negative, M₁ and I₁ must enter the triode region so as to satisfy KCL at node P, introducing large nonlinearity. Since the random baseband signal occasionally assumes large voltage excursions, it is difficult to avoid this effect unless the amount of degeneration (e.g., R_S) is chosen conservatively large, in which case the mixer gain and hence the output swing suffer.

The above observation indicates that the current available to perform upconversion and produce RF swings is approximately equal to the difference between I₁ and I₃ (or between I₂ and I₄). The maximum baseband peak single-ended voltage swing is thus given by

(6.155)-(6.156)

Mixer Carrier Feedthrough

Transmitters using active upconversion mixers potentially exhibit a higher carrier feedthrough than those incorporating passive topologies. This is because, in addition to the baseband DAC offset, the mixers themselves introduce considerable offset. In the circuits of Figs. 6.92 and 6.94(a), for example, the baseband input transistors suffer from mismatches between their threshold voltages and other parameters. Even more pronounced is the offset in the folded mixer of Fig. 6.94(b), as calculated in the following example.

In addition to offset, the six transistors in Fig. 6.96(a) also contribute noise, potentially a problem in GSM transmitters.¹⁴ It is interesting to note that LO duty cycle distortion does not cause carrier feedthrough in double-balanced active mixers. This is studied in Problem 6.15.

Active mixers readily lend themselves to quadrature upconversion because their outputs can be summed in the current domain. Figure 6.97 shows an example employing folded mixers.

Figure 6.97 Summation of quadrature outputs.

Design Procedure

As mentioned in Section 6.1, the design of upconversion mixers typically follows that of the power amplifier. With the input capacitance of the PA (or PA driver) known, the mixer output inductors, e.g., L₁ and L₂ in Fig. 6.97, are designed to resonate at the frequency of interest. At this point, the capacitance contributed by the switching quads, C_q, is unknown and must be guessed. Thus,

(6.161)

where C_L includes the input capacitance of the next stage and the parasitic of L₁ or L₂. Also, the finite Q of the inductors introduces a parallel equivalent resistance given by

(6.162)

If sensing quadrature baseband inputs with a peak single-ended swing of V_a, the circuit of Fig. 6.97 produces an output swing given by

(6.163)

where the factor of results from summation of quadrature signals, 2V_a denotes the peak differential swing at each input, and g_mp is the transconductance of the input PMOS devices. Thus, R_S, g_mp, and V_a must be chosen so as to yield both the required output swing and proper linearity.

How do we choose the bias currents? We must first consider the following example.

The above example suggests that I₀ must be sufficiently large to yield the required output swing. That is, with R_p known, I₀ can be calculated. A double-balanced version of the circuit generates twice the output swing, and a quadrature topology (Fig. 6.97) raises the result by another factor of , delivering a peak output swing of . With I₀ (= I₃/2 = I₄/2 in Fig. 6.97) known, we select I₁ = I₂ = I₃/2 = I₄/2.

How do we select the transistor dimensions? Let us first consider the switching devices, noting that each switching pair in Fig. 6.97 carries a current of nearly I₃ (= I₄) at the extremes of the baseband swings. These transistors must therefore be chosen wide enough to (1) carry a current of I₃ while leaving adequate voltage headroom for I₃ and I₄, and (2) switch their tail currents nearly completely with a given LO swing.

Next, the transistors implementing I₃ and I₄ are sized according to their allowable voltage headroom. Lastly, the dimensions of the input differential pair and the transistors realizing I₁ and I₂ are chosen. With these choices, the input-referred offset [Eq. (6.160)] must be checked.

References

[1] B. Razavi, “A Millimeter-Wave Circuit Technique,” IEEE J. of Solid-State Circuits, vol. 43, pp. 2090–2098, Sept. 2008.

[2] P. Eriksson and H. Tenhunen, “The Noise Figure of A Sampling Mixer: Theory and Measurement,” IEEE Int. Conf. Electronics, Circuits, and Systems, pp. 899–902, Sept. 1999.

[3] S. Zhou and M. C. F. Chang, “A CMOS Passive Mixer with Low Flicker Noise for Low-Power Direct-Conversion Receivers,” IEEE J. of Solid-State Circuits, vol. 40, pp. 1084, 1093, May 2005.

[4] D. Leenaerts and W. Readman-White, “1/f Noise in Passive CMOS Mixers for Low and Zero IF Integrated Receivers,” Proc. ESSCIRC, pp. 41–44, Sept. 2001.

[5] A. Mirzaei et al., “Analysis and Optimization of Current-Driven Passive Mixers in Narrow-band Direct-Conversion Receivers,” IEEE J. of Solid-State Circuits, vol. 44, pp. 2678–2688, Oct. 2009.

[6] D. Kaczman et al., “A Single-Chip 10-Band WCDMA/HSDPA 4-Band GSM/EDGE SAW-less CMOS Receiver with DigRF 3G Interface and +90-dBm IIP2,” IEEE J. Solid-State Circuits, vol. 44, pp. 718–739, March 2009.

[7] H. Darabi and A. A. Abidi, “Noise in RF-CMOS Mixers: A Simple Physical Model,” IEEE J. of Solid-State Circuits, vol. 35, pp. 15–25, Jan. 2000.

[8] W. H. Sansen and R. G. Meyer, “Distortion in Bipolar Transistor Variable-Gain Amplifiers,” IEEE Journal of Solid-State Circuits, vol. 8, pp. 275–282, Aug. 1973.

[9] B. Razavi, “A 60-GHz CMOS Receiver Front-End,” IEEE J. of Solid-State Circuits, vol. 41, pp. 17–22, Jan. 2006.

[10] B. Razavi, “A 900-MHz CMOS Direct-Conversion Receiver,” Dig. of Symposium on VLSI Circuits, pp. 113–114, June 1997.

[11] M. Brandolini et al., “A +78-dBm IIP2 CMOS Direct Downconversion Mixer for Fully-Integrated UMTS Receivers,” IEEE J. Solid-State Circuits, vol. 41, pp. 552–559, March 2006.

[12] D. Manstretta, M. Brandolini, and F. Svelto, “Second-Order Intermodulation Mechanisms in CMOS Downconverters,” IEEE J. Solid-State Circuits, vol. 38, pp. 394–406, March 2003.

[13] H. Darabi and J. Chiu, “A Noise Cancellation Technique in Active RF-CMOS Mixers,” IEEE J. of Solid-State Circuits, vol. 40, pp. 2628–2632, Dec. 2005.

[14] R. S. Pullela, T. Sowlati, and D. Rozenblit, “Low Flicker Noise Quadrature Mixer Topology,” ISSCC Dig. Tech. Papers, pp. 76–77, Feb. 2006.

[15] B. Razavi, “CMOS Transceivers for the 60-GHz Band,” IEEE Radio Frequency Integrated Circuits Symposium, pp. 231–234, June 2006.

Problems

6.1. Suppose in Fig. 6.13, the LNA has a voltage gain of A₀ and the mixers have a high input impedance. If the I and Q outputs are simply added, determine the overall noise figure in terms of the NF of the LNA and the input-referred noise voltage of the mixers.

6.2. Making the same assumptions as in the above problem, determine the noise figure of a Hartley receiver. Neglect the noise of the 90°-phase-shift circuit and the output adder.

6.3. Consider the circuit of Fig. 6.99, where C₁ and C₂ are identical and represent the gate-source capacitances in Fig. 6.15(b). Assume V₁ = −V₂ = V₀ cos ω_LOt.

Figure 6.99 Capacitors driven by differential waveforms.

(a) If C₁ = C₂ = C₀(1 + α₁V), where V denotes the voltage across each capacitor, determine the LO feedthrough component(s) in V_out. Assume α₁V 1.

(b) Repeat part (a) if C₁ = C₂ = C₀(1 + α₁V + α₂V²).

6.4. We express V_n1 in Fig. 6.29(c) as the product of the shaped resistor noise voltage and a square wave toggling between 0 and 1. Prove that the spectrum of V_n1 is given by Eq. (6.31).

6.5. Prove that the voltage conversion gain of a sampling mixer approaches 6 dB as the width of the LO pulses tends to zero (i.e., as the hold time approaches the LO period).

6.6. Consider the LO buffer shown in Fig. 6.55. Prove that the noise of M₅ and M₆ appears differentially at nodes A and B (but the noise due to the loss of the tanks does not).

6.7. In the active mixer of Fig. 6.57, I_n,M1 contains all frequency components. Prove that the convolution of these components with the harmonics of the LO in essence multiplies 4kTγ/g_m by a factor of π²/4.

6.8. If transistors M₂ and M3 in Fig. 6.60(a) have a threshold mismatch of V_OS, determine the output flicker noise due to the flicker noise of I_SS.

6.9. Shown in Fig. 6.100 is the front end of a 1.8-GHz receiver. The LO frequency is chosen to be 900 MHz and the load inductors and capacitances resonate with a quality factor of Q at the IF. Assume M₁ is biased at a current of I₁, and the mixer and the LO are perfectly symmetric.

(a) Assuming M₂ and M₃ switch abruptly and completely, compute the LO-IF feedthrough, i.e., the measured level of the 900-MHz output component in the absence of an RF signal.

(b) Explain why the flicker noise of M₁ is critical here.

Figure 6.100 Front-end chain for a 1.8-GHz RX.

6.10. Suppose the helper in Fig. 6.67 reduces the bias current of the switching pair by a factor of 2. By what factor does the input-referred contribution of the flicker noise fall?

6.11. In the circuit of Fig. 6.67, we place a parallel RLC tank in series with the source of M₄ such that, at resonance, the noise contribution of M₄ is reduced. Recalculate Eq. (6.116) if the tank provides an equivalent parallel resistance of R_p. (Bear in mind that R_p itself produces noise.)

6.12. Can the circuit of Fig. 6.81(a) be viewed as a differential pair whose tail current is modulated at a rate of 2f_LO? Carry out the analysis and explain your result.

6.13. Suppose the quadrature upconversion mixers in a GSM transmitter operate with a peak baseband swing of 0.3 V. If the TX delivers an output power of 1 W, determine the maximum tolerable input-referred noise of the mixers such that the transmitted noise in the GSM RX band does not exceed −155 dBm.

6.14. Prove that the voltage conversion gain of a single-balanced return-to-zero mixer is equal to 2/π even for upconversion.

6.15. Prove that LO duty cycle distortion does not introduce carrier feedthrough in double-balanced active mixers.

6.16. The circuit shown in Fig. 6.101 is a dual-gate mixer used in traditional microwave design. Assume when M₁ is on, it has an on-resistance of R_on1. Also, assume abrupt edges and a 50% duty cycle for the LO and neglect channel-length modulation and body effect.

Figure 6.101 Dual-gate mixer.

(a) Compute the voltage conversion gain of the circuit. Assume M₂ does not enter the triode region and denote its transconductance by g_m2.

(b) If R_on1 is very small, determine the IP₂ of the circuit. Assume M₂ has an overdrive of V_GS0 − V_TH in the absence of signals (when it is on).

6.17. Consider the active mixer shown in Fig. 6.102, where the LO has abrupt edges and a 50% duty cycle. Also, channel-length modulation and body effect are negligible. The load resistors exhibit mismatch, but the circuit is otherwise symmetric. Assume M₁ carries a bias current of I_SS.

(a) Determine the output offset voltage.

(b) Determine the IP₂ of the circuit in terms of the overdrive and bias current of M₁.

Figure 6.102 Active mixer with load mismatch.