## Upsampling and Downsampling

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:

*images*appear around the unit circle. For , this is depicted in Fig.11.2.

### Downsampling (Decimation) Operator

Figure 11.3 shows the symbol for downsampling by the factor . The downsampler selects every th sample and discards the rest:*expanded*by the factor , wrapping times around the unit circle, adding to itself times. For , two partial spectra are summed, as indicated in Fig.11.4. Using the common

*twiddle factor*notation

(12.1) |

the aliasing expression can be written as

#### Example: Downsampling by 2

For , downsampling by 2 can be expressed as , so that (since )#### Example: Upsampling by 2

For , upsampling (stretching) by 2 can be expressed as, so that

(12.2) |

as discussed more fully in §2.3.11.

### Filtering and Downsampling

Because downsampling by causes aliasing of any frequencies in the original signal above , the input signal may need to be first lowpass-filtered to prevent this aliasing, as shown in Fig.11.5. Suppose we implement such an anti-aliasing lowpass filter as an FIR filter of length with cut-off frequency .^{12.1}This is drawn in direct form in Fig.11.6.

*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
(12.3)

is scaled by . - Subphase signal 1
(12.4)

is scaled by , - Subphase signal

(12.5) |

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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Polyphase Decomposition

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