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.

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.

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

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).

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

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 IP_{3} of the mixer is scaled down by the LNA gain. (Recall from Chapter 5 that the mixer noise and IP_{3} are divided by *different* gains.) The design of downconversion mixers therefore entails a compromise between the noise figure and the IP_{3} (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 IP_{2} of the LNA/mixer cascade must be maximized. In Section 6.4, we introduce methods of raising the IP_{2} 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.

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

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.

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.

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).

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.

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 IP_{2}.)

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

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.

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

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.

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.

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

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,

where SNR* _{I}* and SNR

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 Δ*T* seconds and off for *T* − Δ*T* seconds, then the resulting spectrum is still white and its power is scaled by Δ*T*/*T*. This result proves useful in the study of mixers and oscillators.

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.

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

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.

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

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, Δ*T*.

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 3*f _{LO}* is attenuated by about only 10 dB as it appears in the baseband.

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.

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.

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 _{LO}t*. In the frequency domain, this harmonic consists of two impulses at ±

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

We first recall the following Fourier transform pairs:

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 *y*_{1}(*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

where *T _{LO}* denotes the LO period. It follows from Eqs. (6.9) and (6.11) that

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

The impulse, in essence, computes [1/(*jω*)][1 − exp(−*jωT _{LO}*/2)] at ±1/

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

The second output in Fig. 6.21(b), *y*_{2}(*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,

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 *Y*_{1}(*f*) and *Y*_{2}(*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 *Y*_{2}(*f*) is obtained by setting *k* to ±1:

The term in the second set of square brackets must be calculated at the IF. If the IF is much lower than 2*f _{LO}*, then exp(−

Note that *Y*_{2}(*f*) in fact contains a larger IF component than does *Y*_{1}(*f*). The total IF output is therefore equal to

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.

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 *M*_{1}–*M*_{4}, 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.

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

In practice, however, mismatches between *M*_{1} and *M*_{2} 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.

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 4

If we select *R _{on}*

Dividing this result by 1/*π*^{2}, we have

That is, the noise power of *R _{L}* (= 4

The reader may wonder if our choice *R _{on}*

This function reaches a minimum of

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 *R*_{1} denotes the switch resistance), if *V _{in}* = 0,

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 *R*_{1} when *S*_{1} 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 *R*_{1} 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 *R*_{1}, (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 *R*_{1}. Similarly, we use the notation even though its low-frequency components are removed and considered separately.

To calculate the spectrum of *V*_{n1}, we view this waveform as the product of *V _{n,LPF}*(

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

Note that this is the two-sided spectrum of .

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[

where *a* = 2*πR*_{1}*C*_{1}*f _{LO}*. For the summation in Eq. (6.34), we have

Also, typically (2*πR*_{1}*C*_{1})^{−1} *> f _{LO}* and hence coth(2

This result must be multiplied by the sinc^{2} envelope, |(*jω*)^{−1}[1 − exp(−*jωT _{LO}*/2)]|

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

Summing the *one-sided* spectra of *V*_{n1} and *V*_{n2} and the low-frequency contribution, 4*kTR*_{1}, gives the total (one-sided) output noise at the IF:

The input-referred noise is obtained by dividing this result by 1/*π*^{2} + 1/4:

Note that [2] and [3] do not predict the dependence on *R*_{1} or *C*_{1}.

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

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 *R*_{1} 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*,

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

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 *C*_{1}, and the network consisting of *R _{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

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 *S*_{1} is assumed ideal for now. Recall from Fig. 6.21 that the output voltage can be viewed as the sum of two waveforms *y*_{1}(*t*) and *y*_{2}(*t*), given by Eqs. (6.12) and (6.16), respectively. The current drawn by *C*_{1} in Fig. 6.36 is equal to

Moreover, *i _{in}*(

where *Y*(*f*) is equal to the sum of *Y*_{1}(*f*) and *Y*_{2}(*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}*(

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}*)

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

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

The input impedance thus contains a parallel resistive component equal to 1/(2*fC*_{1}) [Fig. 6.37(b)]. Finally, if *ω* 2*πf _{LO}*, the second term is much less than the first, yielding

For the input impedance of a single-balanced mixer, we must add the switch on-resistance, *R*_{1}, to the inverse of Eq. (6.47) and halve the result. If *ω* ≈ *ω _{LO}*, then

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,

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.

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.

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}*(

The input current is routed to the upper arm for 50% of the time and flows through *Z _{BB}*. In the time domain [5],

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,

where *S*(*f*) is the spectrum of a square wave. As expected, upon convolution with the first harmonic of *S*(*f*), *I _{in}*(

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, *V*_{1}(*t*) is multiplied by *S*(*t*) as it returns to the input, and its spectrum is translated to RF. The spectrum of *V*_{2}(*t*) is also upconverted and added to this result.

Figure 6.39(c) summarizes our findings, revealing that the downconverted spectrum of *I _{in}*(

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]:

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.]

Another useful attribute of the 25% duty cycle in Fig. 6.40 is that the mixer switches driven by LO_{0} and LO_{180} (or by LO_{90} and LO_{270}) are not on simultaneously. As a result, the mixer contributes smaller noise and nonlinearity [6].

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}*/

Figure 6.42 depicts a typical single-balanced realization. Here, *M*_{1} converts the input RF voltage to a current (and is hence called a “transconductor”), the differential pair *M*_{2}–*M*_{3} commutates (steers) this current to the left and to the right, and *R*_{1} and *R*_{2} convert the output currents to voltage. We call *M*_{2} and *M*_{3} 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.

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 *X*_{1} and *Y*_{1} are equal to the negative of those at *X*_{2} and *Y*_{2}, respectively. That is, *V*_{X1} = −*V*_{Y1} = −*V*_{X2} = *V*_{Y2}, allowing us to short *X*_{1} to *Y*_{2} and *X*_{2} to *Y*_{1} 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

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.

In the circuit of Fig. 6.42, transistor *M*_{1} 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

Since *V _{out}* =

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/

If *I _{RF}*(

The voltage conversion gain is therefore equal to

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 IP_{3} (Chapter 5). Thus, *V*_{DS1,min} = *V*_{GS1} − *V*_{TH1}. The transconductance of *M*_{1} is limited by the current budget and IP_{3}, as expressed by *g*_{m1} = 2*I*_{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

Suppose the gate voltages of *M*_{2} and *M*_{3} in Fig. 6.42 are held at the common-mode level of the differential LO waveforms, *V _{CM,LO}* [Fig. 6.45(a)]. If

Now consider the time instant at which the gate voltages of *M*_{2} and *M*_{3} reach *V _{CM,LO}* +

which, from Eq. (6.61), reduces to

Thus, *V _{X,min}* must accommodate the overdrive of

Since each resistor carries half of *I*_{D1},

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

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

How much room for improvement do we have? Given by IP_{3} 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 *M*_{2} and *M*_{3} are near equilibrium, the RF current produced by *M*_{1} 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).

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, Δ*T* [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.

A differential pair having an equilibrium overdrive of (*V _{GS}* −

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, *M*_{2} is on and *M*_{3} is off, yielding a total capacitance at node *P* equal to

Note that *C*_{GS3} is substantially smaller than *C*_{GS2} in this phase (why?). The RF current produced by *M*_{1} is split between *C _{P}* and the resistance seen at the source of

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

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.

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,

In this phase, the circuit reduces to a cascode structure, with *M*_{2} 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

Now consider a more realistic case where the LO transitions are not abrupt, allowing *M*_{2} and *M*_{3} 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 *M*_{2} and *M*_{3}—while the noise of *M*_{1} has little effect on the output because it behaves as a common-mode disturbance.

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 *M*_{2} and *M*_{3} 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 *M*_{2} and *M*_{3}. It is therefore necessary to simulate the noise behavior of mixers with the LO circuitry present.

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 *M*_{1} (*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 *M*_{1} and *M*_{2} is injected into node *X*, generating an output noise spectral density given by , where denotes the noise current injected by *M*_{2} into node *X*. The total noise at node *X* is therefore equal to

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

If the effect of *C _{P}* is negligible, then

The term *π*^{2}*kTγ*/*g*_{m1} in (6.85) represents the input-referred contribution of *M*_{1}. This appears puzzling: why is this contribution simply not equal to the gate-referred noise of *M*_{1},4*kTγ*/*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 Δ*T* = 2(*V _{GS}* −

where we assume *g*_{m2} ≈ *g*_{m3}. Now, this noise power must be weighted by a factor of 2Δ*T*/*T _{LO}*, and that in the numerator of Eq. (6.83) by a factor of 1 − 2Δ

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 Δ*T* ∝ (*V _{GS}* −

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

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 *M*_{2} and *M*_{3} 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

obtaining

In the vicinity of *t* = 0, we have

The crossing of LO and is displaced from its ideal point by an amount of Δ*T* (*ω _{LO}* Δ

That is,

Note that 2*V _{p,LO}ω_{LO}* is the slope of the

We now assume nearly abrupt drain current switching for *M*_{2} and *M*_{3} 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 2*I _{SS}* and width Δ

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

The baseband component is obtained for *k* = 0 because *V*_{n2}(*f*) has a low-pass spectrum. It follows that

and hence

In other words, the flicker noise of each transistor is scaled by a factor of *I _{SS}R_{D}*/(

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 *M*_{2} and *M*_{3} as well. Indeed, the analysis is identical [7] and the same results are obtained, with *V*_{n2}(*f*) replaced with 4*kTγ*/*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(

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]

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

The linearity of active mixers is determined primarily by the input transistor’s overdrive voltage. As explained in Chapter 5, the IP_{3} 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

We also noted in Section 6.3.1 that the headroom consumed by the input transistor, *V _{GS}* −

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 *M*_{2} is in the triode region while *M*_{3} 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 *M*_{2} and *M*_{3} operate in saturation, then the division of *I _{RF}* between the two transistors is given by their transconductances and is independent of their

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

where *R _{P}* denotes the “average resistance” seen at the common source node of

The IP_{2} of active mixers is also of great interest. We compute the IP_{2} in Section 6.4.3.

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.

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} = 2*I*_{0} and each current source carries a fraction, *αI*_{0}, then *R _{D}* can be as large as

But how about the noise contributed by *M*_{4} and *M*_{5}? Assuming that these devices are biased at the edge of saturation, i.e., |*V _{GS}* −

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

Interestingly, the total noise due to each current-source helper and its corresponding load resistor *rises* with *α*, beginning from 4*kTI*_{0}/*V*_{0} for *α* = 0 and reaching (4*kTI*_{0}/*V*_{0})(2*γ*) 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.

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 *M*_{1} by *M*_{4}, thereby reducing the current flowing through the load resistors (and the switching transistors). For example, if |*I*_{D4}| = 0.75*I*_{D1}, then *R _{D}* and hence the gain can be quadrupled. Moreover, the reduction of the bias current switched by

The above approach nonetheless faces two issues. First, transistor *M*_{4} contributes additional capacitance to node *P*, exacerbating the difficulties mentioned earlier. As a smaller bias current is allocated to *M*_{2} and *M*_{3}, raising the impedance seen at their source [≈ 1/(2*g _{m}*)],

In order to suppress the capacitance and noise contribution of *M*_{4} 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

In the circuit of Fig. 6.69(a), the inductor parasitics must be managed carefully. First, *L*_{1} contributes some capacitance to node *P*, equivalently raising *C _{P}*. Second, the loss of

where *C _{P,tot}* includes the capacitance of

The circuits of Figs. 6.67 and 6.69 suffer from a drawback in deep-submicron technologies: since *M*_{1} is typically a small transistor, it poorly matches the current mirror arrangement that feeds *M*_{4}. 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, *C*_{1} acts as a short circuit at RF and *L*_{1} resonates with the parasitics at nodes *P* and *N*. Furthermore, the voltage headroom available to *M*_{1} is no longer constrained by (*V _{GS}* −

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-IP_{2} mixers.

It is instructive to compute the IP_{2} of a single-balanced mixer in the presence of asymmetries. (Recall from Chapter 4 that a symmetric mixer has an infinite IP_{2}.) Let us begin with the circuit of Fig. 6.71(a), where *V _{OS}* denotes the offset voltage associated with

As shown in Fig. 6.71(b), the vertical shift of *V _{LO}* displaces the consecutive crossings of LO and by ± Δ

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

where *V*_{GS0} is the bias gate-source voltage of *M*_{1}. With a square-law device, the IM_{2} product emerges in the current of *M*_{1} as

Multiplying this quantity by *V _{OS}R_{D}*/(

To calculate the IP_{2}, the value of *V _{m}* must be raised until the amplitude of

Writing *g*_{m1} as *μ _{n}C_{ox}*(

For example, if (*V _{GS}* −

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 − Δ

Equating the amplitude of this component to (2/*π*)*g _{m}*

For example, a duty cycle of 48% along with (*V _{GS}* −

In order to raise the IP_{2}, 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 IM_{2} product only as a result of *mismatches* between *M*_{1} and *M*_{2}. We quantify this effect in the following example. Note that, unlike the previous double-balanced mixers, this circuit employs a tail current source.

While improving the IP_{2} significantly, the use of a differential pair in Fig. 6.72 *degrades* the IP_{3}. As formulated in Chapter 5, a quasi-differential pair (with the sources held at ac ground) exhibits a higher IP_{3}. 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

While *independent* of *V*_{OS1}, the low-frequency beat in *I*_{D1} is multiplied by a factor of *V*_{OS2}*R _{D}*/(

Noting that the output amplitude of each fundamental is equal to (2/*π*)2*V _{m}g_{m}*

For example, if *V _{GS}* −

We have thus far considered one mechanism leading to a finite IP_{2}: 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

While conceived for noise and gain optimization reasons, the mixer topology in Fig. 6.70 also exhibits a high IP_{2}. The high-pass filter consisting of *L*_{1}, *C*_{1}, and the resistance seen at node *P* suppresses low-frequency beats generated by the even-order distortion in *M*_{1}. From the equivalent circuit shown in Fig. 6.74, we have

At low frequencies, this result can be approximated as

revealing a high attenuation.

Another approach to raising the IP_{2} 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

we recognize that the gain at low frequencies falls in proportion to *C _{d}s*, making

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 IP_{2} (if the offset of the switching pair is considered). We therefore expect a higher IP_{2} if an inductor resonates with this capacitance. Figure 6.77 shows a double-balanced mixer employing both capacitive degeneration and resonance to achieve an IP_{2} of + 78 dBm [11].

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 IP_{2} of active mixers, we postulate that the techniques described in Section 6.4.3 for raising the IP_{2} 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 2*ω _{LO}*, the diode-connected transistor turns on when LO and cross and

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

The circuit of Fig. 6.79 nonetheless requires large LO swings to ensure that *V _{P}* and

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 *S*_{1} is driven by a waveform having a frequency of 2*f _{LO}* but a duty cycle of, say, 80%. As depicted in Fig. 6.81(b),

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 2*f _{LO}* with such narrow pulses; the conversion gain thus suffers because the transconductor remains off for a greater portion of the period. Third, switch

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.

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.

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.

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, *y*_{1}(*t*) and *y*_{2}(*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 *Y*_{1}(*f*) still occurs at *k* = ± 1 and is given by

For *Y*_{2}(*f*), we must also set *k* to ±1:

However, the term in the second set of brackets must be evaluated at the *upconverted* frequency. If *ω* = *ω _{LO}* +

indicating that the upconverted output amplitude is proportional to *ω _{BB}*/(

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.

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 (

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 *L*_{1} resonates with the total capacitance at *X* and *Y*, and its value is chosen to yield

where *C _{X,Y}* denotes the capacitances contributed by the switches at

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 *M*_{1}–*M*_{2} and *M*_{3}–*M*_{4} 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 *M*_{1}–*M*_{4} also yield a relatively high linearity.^{12}

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.

The third issue concerns the available overdrive voltage of the mixer switches, a particularly serious problem in Fig. 6.85(b). We note that *M*_{5} can be ac coupled to *M*_{1}, 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

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

Now, the dc level of the LO is approximately equal to *V _{DD}*, with the peak reaching

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

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 *M*_{5} and biased at *V _{DD}*.

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

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

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

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/

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.

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

where *C _{X}* denotes the total capacitance seen from

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

where *R _{p}* is the equivalent parallel resistance of each inductor at resonance.

With only low frequencies present at the gates and drains of *M*_{1} and *M*_{2} 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

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

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)].

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 *I*_{3}, starving *M*_{3} and *M*_{5}. Now, if the differential input becomes more negative, *M*_{1} and *I*_{1} 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 *I*_{1} and *I*_{3} (or between *I*_{2} and *I*_{4}). The maximum baseband peak single-ended voltage swing is thus given by

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.

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., *L*_{1} and *L*_{2} 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,

where *C _{L}* includes the input capacitance of the next stage and the parasitic of

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

where the factor of results from summation of quadrature signals, 2*V _{a}* denotes the peak differential swing at each input, and

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

The above example suggests that *I*_{0} must be sufficiently large to yield the required output swing. That is, with *R _{p}* known,

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 *I*_{3} (= *I*_{4}) at the extremes of the baseband swings. These transistors must therefore be chosen wide enough to (1) carry a current of *I*_{3} while leaving adequate voltage headroom for *I*_{3} and *I*_{4}, and (2) switch their tail currents nearly completely with a given LO swing.

Next, the transistors implementing *I*_{3} and *I*_{4} are sized according to their allowable voltage headroom. Lastly, the dimensions of the input differential pair and the transistors realizing *I*_{1} and *I*_{2} are chosen. With these choices, the input-referred offset [Eq. (6.160)] must be checked.

[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.

6.1. Suppose in Fig. 6.13, the LNA has a voltage gain of *A*_{0} 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 *C*_{1} and *C*_{2} are identical and represent the gate-source capacitances in Fig. 6.15(b). Assume *V*_{1} = −*V*_{2} = *V*_{0} cos *ω _{LO}t*.

(a) If *C*_{1} = *C*_{2} = *C*_{0}(1 + *α*_{1}*V*), where *V* denotes the voltage across each capacitor, determine the LO feedthrough component(s) in *V _{out}*. Assume

(b) Repeat part (a) if *C*_{1} = *C*_{2} = *C*_{0}(1 + *α*_{1}*V* + *α*_{2}*V*^{2}).

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 *M*_{5} and *M*_{6} 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 4*kTγ*/*g _{m}* by a factor of

6.8. If transistors *M*_{2} and *M*3 in Fig. 6.60(a) have a threshold mismatch of *V _{OS}*, determine the output flicker noise due to the flicker noise of

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 *M*_{1} is biased at a current of *I*_{1}, and the mixer and the LO are perfectly symmetric.

(a) Assuming *M*_{2} and *M*_{3} 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 *M*_{1} is critical here.

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 *M*_{4} such that, at resonance, the noise contribution of *M*_{4} is reduced. Recalculate Eq. (6.116) if the tank provides an equivalent parallel resistance of *R _{p}*. (Bear in mind that

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 2*f _{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 *M*_{1} 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.

(a) Compute the voltage conversion gain of the circuit. Assume *M*_{2} does not enter the triode region and denote its transconductance by *g*_{m2}.

(b) If *R*_{on1} is very small, determine the *IP*_{2} of the circuit. Assume *M*_{2} 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 *M*_{1} carries a bias current of *I _{SS}*.

(a) Determine the output offset voltage.

(b) Determine the *IP*_{2} of the circuit in terms of the overdrive and bias current of *M*_{1}.

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