13.19. THE ZOOM FFT
The Zoom FFT is interesting because it blends
complex downconversion, lowpass filtering, and sample rate change
through decimation in a spectrum analysis application. The Zoom FFT
method of spectrum analysis is used when fine spectral resolution
is needed within a small portion of a signal's overall frequency
range. This technique is more efficient than the traditional FFT in
such a situation.
Think of the spectral analysis situation where
we require fine frequency
resolution, closely spaced FFT bins, over the frequency range
occupied by the signal of interest shown in Figure 1352(a). (The other signals are of no
interest to us.) We could collect many time samples and perform a
largesize radix2 FFT to satisfy our fine spectral resolution
requirement. This solution is inefficient because we'd be
discarding most of our FFT results. The Zoom FFT can help us
improve our computational efficiency through:
The process begins with the continuous x(t)
signal being digitized at a sample rate of f_{s}_{1} by an
analogtodigital (A/D) converter yielding the Npoint x(n)
time sequence whose spectral magnitude is X(m)
in Figure 1352(a). The
Zoom FFT technique requires narrowband filtering and decimation in
order to reduce the number of time samples prior to the final FFT,
as shown in Figure
1352(b). The downconverted signal's spectrum, centered at zero
Hz, is the X_{c}(m) shown in Figure 1352(c). (The lowpass filter's
frequency response is the dashed curve.) After lowpass filtering
x_{c}(n), the filter's output is decimated by
an integer factor D yielding a
time sequence whose
sample rate is f_{s}_{2} = f_{s}_{1}/D prior to the FFT operation. The key
here is that the length of is
N/D, allowing a reducedsize FFT. (N/D
must be an integer power of two to enable the use of radix2 FFTs.)
We perform the FFT only over the decimated signal's bandwidth. It's
of interest to note that, because its input is complex, the N/Dpoint FFT has a nonredundant
frequency analysis range from –f_{s}_{2}/2 to +f_{s}_{2}/2. (Unlike the
case of real inputs, where the positive and negative frequency
ranges are redundant.)
The implementation of the Zoom FFT is given in
Figure 1353, where all
discrete sequences are real valued.
Figure 1353. Zoom FFT processing
details.
Relating the discrete sequences in Figure 1352(b) and Figure 1353, the complex
time sequence x_{c}(n) is represented mathematically as:
while the complex decimated sequence is
Equation 13101
The complex mixing sequence , where
t_{s}_{1} =
1/f_{s}_{1}, can
be represented in the two forms of
Equation 13102
Let's use an example to illustrate the Zoom
FFT's ability to reduce standard FFT computational workloads. Say
we have a signal sampled at f_{s}_{1} = 1024 kHz,
and we require 2 kHz FFT bin spacing for our spectrum analysis
resolution. These conditions require us to collect 512 samples
(1024 kHz/2 kHz) and perform a 512point FFT. If the signal of
interest's bandwidth is less than f_{s}_{1}/8, we could
lowpass filter, use a decimation factor of D = 8, and only perform a 512/8 =
64point FFT and still achieve the desired 2 kHz frequency
resolution. We've reduced the 512point FFT (requiring 2304 complex
multiplies) to a 64point FFT (requiring 192 complex multiplies).
That's an computational workload reduction of more than 90%!
We're using the following definition for the
percent computation reduction in complex multiplies afforded by the
(N/D)point Zoom FFT over the standard
Npoint FFT,
Equation 13103
Plotting Eq. (13103) for various decimation factors
over a range of FFT sizes (all integer powers of two) yields the
percent of FFT computational reduction curves in Figure 1354.
Figure 1354. Percent computational
workload reduction, in complex multiplies, of an (N/D)point Zoom FFT relative to a standard
Npoint FFT.
We see that the Zoom FFT affords significant
computational saving over a straightforward FFT for spectrum
analysis of a narrowband portion of some X(m)
spectrum—and the computational savings in complex multiplies
improves as the decimation factor D increases. Ah, but here's the rub. As
D increases, the Zoom FFT's
lowpass filters must become more narrow which increases their
computational workload; this is the tradeoff. What you have to ask
yourself is: "Does the FFT size reduction compensate for the
additional quadrature mixing and dual filtering computational
workload?" It certainly would if a largesize FFT is impossible
with your available FFT hardware or software. If you can arrange for your continuous signal of
interest to be centered at f_{s}_{1}/4, then the
quadrature mixing can be performed without multiplications. (See Section
13.1.) You may be able to use a simple, efficient, IIR filter
if spectral phase is unimportant. If phase distortion is
unacceptable, then efficient polyphase and halfband FIR filters
are applicable. The computationally efficient frequency sampling
and interpolated FIR filters, in Chapter 7, should be
considered. If the signal of interest is very narrowband relative
to the f_{s}_{1}
sample rate, requiring a large decimation factor and very
narrowband computationally expensive filters, perhaps a cascaded
integratorcomb (CIC) filter can be used to reduce the filtering
computational workload.
