Note 65. Generating I and Q Channels Digitally: Rader’s Approach
This note presents an approach for generating inphase and and quadrature channels from a digitized, real-valued bandpass signal.
Radar and communications systems often involve quadrature receivers, which typically exhibit mismatch or imbalances between the amplitude and phase responses of their inphase and quadrature output channels. These imbalances introduce distortion into the signals of interest that are derived from the I and Q outputs of the receiver. Rader’s approach is one of several digital I/Q techniques that have been devised to form the I and Q signals directly from a single real bandpass signal and thereby avoid the I/Q matching limitations of analog quadrature receivers.
65.1. The Approach
Recipe 65.1 implements a digital I/Q approach depicted in Figure 65.1 and originally described by Rader in [1, 2]. The analytic signal generation (ASG) filter shown in the diagram can be implemented using any of the methods described in Notes 60 through 64. However, in [1], Rader initially implements this filter as a phase-splitting network consisting of a pair of low-order IIR filters, as shown in Figure 65.3. The top branch, which generates the real part of 
, implements the transfer function
65.1
Figure 65.1. Block diagram of processing steps in Rader’s digital I/Q approach
Recipe 65.1. Generating Digital I and Q
A block diagram of this processing sequence is shown in Figure 65.1. The RF signal, x(t), is assumed to have a bandwidth of B Hz and a carrier frequency of fc > B, as depicted in Figure 65.2.
- The multiplier and bandpass filter form a mixer that translates the signal down to an intermediate frequency of B Hz to yield the signal spectrum shown in Figure 65.2(b).
- The ADC samples the signal at a rate of 4B samples per second to yield a discrete-time signal with the spectrum shown in Figure 65.2(c).
- The ASG filter forms the analytic associate of xs[n] by eliminating the negative frequency sideband to yield the result shown in Figure 65.2(d).
- Downsampling by a factor of 4 causes the positive-frequency sideband to be shifted to a center frequency of zero, as shown in Figure 65.2(e). The real part of the corresponding complex signal is the inphase output, and the imaginary part is the quadrature output.
Figure 65.2. Spectra of signals at various points in Rader’s digital I/Q approach: (a) spectrum of bandpass RF input signal, x(t), with a bandwidth of B and a carrier frequency of fc, (b) spectrum of IF signal, xIF(t), produced by the mixer, (c) baseband image of spectrum for xs[n] created from xIF(t) by sampling at a rate of fs samples per second, (d) spectrum of the analytic signal, , formed by complex filtering of xs[n], (e) spectrum of final quadrature baseband signal after down sampling by a factor of 4. The hashed areas indicate frequencies outside the bandwidth supported by the sampling rate.
The bottom branch, which generates the imaginary part of , implements the transfer function
65.2
The coefficients a2 and b2 can be obtained using the design procedure presented in Note 63. Rader points out that because the filter output, , is destined to be down sampled by a factor of 4, it is possible to build a more efficient filter that absorbs the downsampling operation and generates only every fourth output. This improved realization is shown in Figure 65.4. The down sampling is accomplished by the switches on the output lines.
Careful examination of Figure 65.3 reveals that even-indexed samples of Re
depend only upon odd-indexed samples of , and even-indexed samples of Im depend only upon even-indexed samples of . The input switch shown in Figure 65.4 routes odd-indexed samples to the top branch and even-indexed samples to the bottom branch, thereby enabling each branch to run at half the original sample rate. Because of the reduced rate, the z–2 delays become z–1delays. The z–1 delay in the top branch of Figure 65.3 is accomplished by the relative phasing of the output switch closures in Figure 65.4. To establish the proper phasing, the sample of Q[n′] that is generated when n = 0 is discarded so that I[0] is generated when n = 1, and Q[0] is generated when n = 2.
Figure 65.3. IIR realization for the complex filter
Figure 65.4. Improved IIR realization for the complex filter
Normally, shifting a bandpass spectrum such as Figure 65.2(d) to create a baseband spectrum such as Figure 65.2(e) would require some sort of mixing operation. However, the particular combination of center frequency and sampling rate used in Rader’s approach completely eliminates the need for a final mixing step. If the signal corresponding to the spectrum in Figure 65.5(a) is downsampled by a factor of 4, two things happen:
- Images of the spectrum are created at intervals of the new sampling rate,
, as shown in Figure 65.5(b). - The supported passband of the system changes from ±fS/2 to
, with exactly one image of the signal’s spectrum falling in this new passband, as shown in Figure 65.5(c).
Figure 65.5. When the signal having the spectrum in (a) is downsampled by a factor of 4, images are created at intervals of fs/4, as in (b), and the supported passband shrinks, as depicted in (c), from ±fs/2 to ±f′s/2 where f′s = fs/4.
References
1. C. M. Rader, “A Simple Method for Sampling In-Phase and Quadrature Components,” IEEE Trans. Aerospace and Electronic Syst., vol. 20, no. 6, Nov. 1984, pp. 821–824.
2. C. M. Rader, “Method and Apparatus for Sampling In-Phase and Quadrature Components,” U. S. Patent 4,794,556, dated Dec. 27, 1988.
3. B. Gold and C. M. Rader, Digital Processing of Signals, McGraw-Hill, 1969.