For the DTFT, we proved in Chapter 2 (p. p. ) the stretch theorem (repeat theorem) which relates upsampling (``stretch'') to spectral copies (``images'') in the DTFT context; this is the discrete-time counterpart of the scaling theorem for continuous-time Fourier transforms (§B.4). Also, §2.3.12 discusses the downsampling theorem (aliasing theorem) for DTFTs which relates downsampling to aliasing for discrete-time signals. In this section, we review the main results.
Upsampling (Stretch) Operator
Figure 11.1 shows the graphical symbol for a digital upsampler by the factor . To upsample by the integer factor , we simply insert zeros between and for all . In other words, the upsampler implements the stretch operator defined in §2.3.9:
Downsampling (Decimation) Operator
Figure 11.3 shows the symbol for downsampling by the factor . The downsampler selects every th sample and discards the rest:
In the frequency domain, we have
Using the common twiddle factor notation
the aliasing expression can be written as
For , downsampling by 2 can be expressed as , so that (since )
, upsampling (stretching) by 2 can be expressed as
, so that
as discussed more fully in §2.3.11.
We do not need out of every filter output samples due to the : downsampler. To realize this savings, we can commute the downsampler through the adders inside the FIR filter to obtain the result shown in Fig.11.7. The multipliers are now running at times the sampling frequency of the input signal . This reduces the computation requirements by a factor of . The downsampler outputs in Fig.11.7 are called polyphase signals. The overall system is a summed polyphase filter bank in which each ``subphase filter'' is a constant scale factor . As we will see, more general subphase filters can be used to implement time-domain aliasing as needed for Portnoff windows (§9.7).
We may describe the polyphase processing in the anti-aliasing filter of Fig.11.7 as follows:
- Subphase signal 0
is scaled by .
- Subphase signal 1
is scaled by ,
- Subphase signal
which we recognize as a direct-form-convolution implementation of a length FIR filter , with its output downsampled by the factor .
The summed polyphase signals of Fig.11.7 can be interpreted as ``serial to parallel conversion'' from an ``interleaved'' stream of scalar samples to a ``deinterleaved'' sequence of buffers (each length ) every samples, followed by an inner product of each buffer with . The same operation may be visualized as a deinterleaving through variable gains into a running sum, as shown in Fig.11.8.
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