In digital communications applications, the FIR implementation of the HT (such as that in Figure 912(b)) is used to generate a complex analytic signal xc(n). Some practitioners now use a timedomain complex filtering technique to achieve analytic signal generation when dealing with real bandpass signals[8]. This scheme, which does not specifically perform the HT of an input sequence xr(n), uses a complex filter implemented with two real FIR filters with essentially equal magnitude responses, but whose phase responses differ by exactly 90o, as shown in Figure 915.
Figure 915. Generating an xc(n) sequence with a complex filter (two real FIR filters).
Here's how it's done. A standard Ktap FIR lowpass filter is designed, using your favorite FIR design software, to have a twosided bandwidth slightly wider than the original real bandpass signal of interest. The real coefficients of the lowpass filter, hLP(k), are then multiplied by the complex exponential , using the following definitions:
wo 
= 
center frequency, in radians/second, of original bandpass signal, (wo = 2pfo) 
fo 
= 
center frequency, in Hz 
n 
= 
time index of the lowpass filter coefficients (n = 0,1,2,...K–1) 
ts 
= 
time between samples, measured in seconds (ts = 1/fs) 
fs 
= 
sample rate of the original xr(n) bandpass signal sequence. 
The results of the multiplication are complex coefficients whose rectangularform representation is
creating a complex bandpass filter centered at fo Hz.
Next we use the real and imaginary parts of the filter's hBP(k) coefficients in two separate realvalued coefficient FIR filters as shown in Figure 915. In DSP parlance, the filter producing the xI(n) sequence is called the Ichannel for inphase, and the filter generating xQ(n) is called the Qchannel for quadrature phase. There are several interesting aspects of this mixing analytic signal generation scheme in Figure 915:
Keep in mind now, the xQ(n) sequence in Figure 915 is not the Hilbert transform of the xr(n) input. That wasn't our goal here. Our intent was to generate an analytic xc(n) sequence whose xQ(n) quadrature component is the Hilbert transform of the xI(n) inphase sequence.
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