Generating Complex Baseband and Analytic Bandpass Signals
There are so many different time and frequencydomain methods for generating complex baseband and analytic bandpass signals that I had trouble keeping those techniques straight in my mind. Thus, for my own benefit, I created a kind of reference table showing those methods. I present that table for your viewing pleasure in this blog.
For clarity, I define a complex baseband signal as follows: derived from an input analog x_{bp}(t)bandpass signal whose spectrum is shown in Figure 1(a), or discrete input x_{bp}(n) bandpass signal whose spectrum is shown in Figure 1(b), a complex baseband signal is an x_{BB}(n) sequence whose spectrum is that shown in Figure 1(c). The sample rate of an x_{bp}(n) input sequence is defined as f_{s} Hz.
Figure 1.
Based on the same analog x_{bp}(t) or discrete x_{bp}(n) input bandpass signal, an analytic bandpass signal is an x_{ABP}(n) sequence whose spectrum is that shown in Figure 1(d).
I realize that, by strict definition, an analytic signal has no negativefrequency spectral energy. And because our x_{ABP}(n) output bandpass signal is a discrete sequence it has spectral replications in its negativefrequency spectral region—so calling x_{ABP}(n) an analytic signal seems incorrect. We'll bypass that controversy by saying that a discrete sequence is analytic if it has no spectral energy in the frequency range of –f_{s}/2 to zero Hz.
Table 1, below, presents my Hit Parade of complex baseband and analytic bandpass signal generation methods. In that table "LPF" means a lowpass, linearphase, tappeddelay line, FIR filter. All discrete Fourier transforms (DFTs) are implemented with radix2 fast Fourier transforms (FFTs).
Table 1: Complex baseband and analytic bandpass signal generation methods
Process 
Input/Output 
Comments 
Quadrature Sampling

Input: analog bandpass signal centered at f_{c} Hz, with sample rate of f_{s} Hz..
Output: discrete complex x_{BB}(n) baseband signal, centered at zero Hz, with sample rate of f_{s} Hz. 
Uses analog mixing and analog lowpass filters. Difficult to control the exact phase delays and gains of the i(t) and q(t) signals. 
Quadrature Sampling

Input: analog bandpass signal centered at f_{c} Hz, with sample rate of f_{s} Hz.
Output: discrete complex x_{BB}(n) baseband signal, centered at zero Hz, with sample rate of f_{s} Hz. 
Alldigital downconversion facilitates exact control of phases and gains of the x_{I}(n) and x_{Q}(n) signals. A/D's f_{s} sample rate normally equal to 4f_{c}, but setting f_{s} = 0.8f_{c} allows bandpass sampling to reduce the f_{s} sample rate. f_{s} must greater than twice the bandwidth of x_{bp}(t). See [1] or Section 8.9 of [2]. 
Discrete Complex Downconversion

Input: discrete real bandpass signal centered at f_{s}/4 Hz, with sample rate of f_{s} Hz.
Output: discrete complex x_{BB}(n) baseband signal, centered at zero Hz, with sample rate of f_{s}/4 Hz. 
Uses a timedomain Hilbert transformer and a halfband highpass FIR filter. If h_{hilb}(k) has an odd number of taps, then half its coefficients will be zerovalued. See [3] or Section 13.43 of [2]. 
Discrete Complex Downconversion

Input: discrete real bandpass signal centered at f_{s}/4 Hz, with sample rate of f_{s} Hz.
Output: discrete complex x_{BB}(n) baseband signal, centered at zero Hz, with sample rate of f_{s}/4 Hz. 
Uses a timedomain Hilbert transformer and simple threetap highpass FIR compensation filter as shown in Part (a) of figure. Efficient implementation shown in Part (b) of figure. For reasonably acceptable operation, the bandwidth of x_{bp}(n) must not be larger than, say, f_{s}/10. See Section 13.43 of [2]. 
Discrete Complex Downconversion

Input: discrete real bandpass signal centered at f_{s}/4 Hz, with sample rate of f_{s} Hz..
Output: discrete complex x_{BB}(n) baseband signal, centered at zero Hz, with sample rate of f_{s}/2 Hz. 
Standard complex downconversion and lowpass FIR filtering, with decimation by two as shown in Part (a) of figure. Due to the decimation, a very efficient implementation is that shown in Part (b) of figure. The coefficients of the Inphase and Quadrature phase LPFs are decimated versions of the coefficients in the identical Part (a) LPFs. If the Part (a) LPFs are halfband filters, the Part (b) Inphase and Quadrature phase LPFs will be even more computationally efficient. See [4], [5], or Section 13.1.3 of [2]. 
Discrete Complex Downconversion

Input: discrete real bandpass signal centered at f_{c} Hz, with sample rate of f_{s} Hz.
Output: discrete complex x_{BB}(n) baseband signal, centered at zero Hz, with sample rate of f_{s} Hz. 
Negativefrequency components of x_{bp}(n) are attenuated in the frequency domain. Frequency downconversion performed by shifting (rotating) the frequencydomain indices of the positivefrequency spectral samples prior to inverse DFT. 
Discrete Complex Downconversion

Input: discrete real bandpass signal centered at kf_{s}/D Hz, with sample rate of f_{s} Hz.
Output: discrete complex x_{BB}(n) baseband signal, centered at zero Hz, with sample rate of f_{s}/D Hz. 
Negativefrequency components of x_{bp}(n) are attenuated in the frequency domain. Frequency downconversion performed by decimation in the time domain. See Section 13.29 of [2].

Analytic Bandpass Signal Generation

Input: discrete bandpass signal centered at f_{c} Hz, with sample rate of f_{s} Hz.
Output: discrete analytic x_{ABP}(n) bandpass signal, centered at f_{c} Hz, with sample rate of f_{s} Hz. 
Standard timedomain, tappeddelay line, FIR Hilbert transform method of discrete analytic bandpass signal generation. For proper time synchronization of x_{I}(n) and x_{Q}(n), N must be an odd number—in which case half the h_{hilb}(k) coefficients will be zerovalued. See Note 61 of [7] or Section 9.4.1 of [2]. 
Analytic Bandpass Signal Generation
Windowing the two sets of filter coefficients will minimize passband magnitude differences between the h_{cos}(k) and h_{sin}(k) filters. (This method has slightly improved aggregate (overall) negative frequency attenuation compared to the following A(k) and B(k) dual filter method.) 
Input: discrete bandpass signal centered at f_{c} Hz, with sample rate of f_{s} Hz.
Output: discrete analytic x_{ABP}(n) bandpass signal, centered at f_{c} Hz, with sample rate of f_{s} Hz. 
A prototype lowpass filter (LPF) is designed to have a twosided bandwidth slightly greater than the bandwidth of x_{bp}(n). The LPF's coefficients are multiplied by cosine and sine sequences whose frequencies are f_{c} Hz, creating a positivefrequency complex h_{bp}(k) bandpass filter. If f_{c} = f_{s}/4 then half the LPF coefficients will be zerovalued. Using a halfband LPF, if possible, further enhances computational efficiency. See [6], Note 62 of [7], or Section 9.5 of [2]. 
Analytic Bandpass Signal Generation
The A(k) and B(k) filters have guaranteed equal magnitude responses. B[k] coefficients are a reversedordered version of the A[k] coefficients (reduces coefficient storage requirement). 
Input: discrete bandpass signal centered at f_{c} Hz, with sample rate of f_{s} Hz.
Output: discrete analytic x_{ABP}(n) bandpass signal, centered at f_{c} Hz, with sample rate of f_{s} Hz. 
Clay Turner's method of designing two orthogonal real bandpass filters, that when combined yield a positivefrequency complex filter. If filters are centered at f_{s}/4 and have an even number of taps, then half the A[k] and B[k] coefficients will be zerovalued. Phase responses of A[k] and B[k] filters are very nearly, but not quite exactly, linear. See [8] for equations used to compute A(k) and B(k) coefficients. 
Analytic Bandpass Signal Generation

Input: discrete bandpass signal centered at f_{c} Hz, with sample rate of f_{s} Hz.
Output: discrete analytic x_{ABP}(n) bandpass signal, centered at f_{c} Hz, with sample rate of f_{s} Hz. 
Straightforward frequencydomain method of bandpass analytic signal generation. Negativefrequency spectral components of X_{bp}(m) are set to zero creating the desired analytic signal's spectrum. See [9] or Section 9.4.2 of [2]. 
Interpolated Analytic Bandpass Signal Generation _{}

Input: discrete bandpass signal centered at f_{c} Hz, with sample rate of f_{s} Hz.
Output: interpolated discrete x_{ABP}(n) analytic bandpass signal, centered at f_{c} Hz, with sample rate of Mf_{s} Hz. 
Frequencydomain method of interpolated by M bandpass analytic signal generation. Negativefrequency spectral components of X_{bp}(m) are set to zero creating the desired analytic signal's spectrum. New spectrum is zerostuffed, prior to inverse DFT, to achieve timedomain interpolation by factor M. See Section 13.28.2 of [2]. 
For completeness, I mention that DSP pioneer Charles Rader proposed a computationallyefficient analytic bandpass signal generation method (See [10] or Note 66 in [7]) where both its x_{I}(n) and x_{Q}(n) output channels have identical frequency magnitude responses, however that scheme does not exhibit a linearphase frequency response. As such, I didn't include it in Table 1.
References
[1] Considine, V. “Digital Complex Sampling,” Electronics Letters, 19, August 4, 1983.
[2] Lyons, R. Understanding Digital Signal Processing, Prentice Hall Publishing, Hoboken, NJ, 2010.
[3] Ohlsson, H., et al. “Design of a Digital Down Converter Using High Speed Digital Filters,” in Proc. Symp. on Gigahertz Electronics, Gothenburg, Sweden, March 13–14, 2000, pp. 309–312.
[4] Powell, S. “Design and Implementation Issues of All Digital Broadband Modems,” DSP World Workshop Proceedings, Toronto, Canada, September 13–16, 1998, pp. 127–142.
[5] Frerking, M. Digital Signal Processing in Communications Systems, Chapman & Hall, New York, 1994, p. 330.
[6] Reilly, A., et. al. “Analytic Signal Generation—Tips and Traps,” IEEE Trans. on Signal Proc., Vol. 42, No. 11, Nov. 1994.
[7] Rorabaugh, C. Notes on Digital Signal Processing: Practical Recipes for Design, Analysis and Implementation, Prentice Hall Publishing, Hoboken, NJ, 2010.
[8] Turner, C. "An Efficient Analytic Signal Generator", IEEE Signal Processing Magazine, DSP Tips & Tricks column, July 2009.
[9] Marple, S., Jr. “Computing the DiscreteTime ‘Analytic’ Signal via FFT,” IEEE Trans. on Signal Proc., Vol. 47, No. 9, September 1999, pp. 2600–2603.
[10] Rader, C. “A Simple Method for Sampling InPhase and Quadrature Components,” IEEE Trans. Aerospace and Electronic Syst., Vol. 20, No. 6, pp. 821824, Nov. 1984.
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