Stretch Operator

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Index: Spectral Audio Signal Processing

Spectral Audio Signal Processing

Fourier Theorems for the DTFT

Power Theorem

Repeat (Scaling) Operator

A filter in the audio signal processing context is any operation that accepts a signal as an input and produces a signal as an output. Most practical audio filters are linear and time invariant, in which case they can be characterized by their impulse response or their frequency response. — Click for https://ccrma.stanford.edu/~jos/filters/What_Filter.html

Sampling is the process of converting a continuous-time signal into a discrete-time signal. — Click for http://ccrma.stanford.edu/~jos/mdft/Sampling_Theory.html

Bandlimited interpolation is the ideal type of interpolation for properly sampled signals. — Click for https://ccrma.stanford.edu/~jos/resample/

The sampling rate of a discrete-time signal is defined as the number of samples per second. Its units are thus in Hertz (Hz). Shannon's Sampling Theorem states that the original continuous-time signal can be recovered exactly from the samples if and only if the sampling rate is higher than twice the highest frequency present in the original signal. Any higher frequencies will alias to frequencies below half the sampling rate. — Click for https://ccrma.stanford.edu/~jos/mdft/Sampling_Theory.html

To downsample a signal by the factor N means to form a new signal consisting of every Nth sample of the original signal. — Click for https://ccrma.stanford.edu/~jos/mdft/Downsampling_Operator.html

Click for https://ccrma.stanford.edu/~jos/sasp/Multirate_Filter_Banks.html

A signal is typically a real-valued function of time. A discrete-time signal is typically a real-valued function of discrete time, and is therefore a time-ordered sequence of real numbers. — Click for http://ccrma.stanford.edu/~jos/filters/Definition_Signal.html

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Index: Spectral Audio Signal Processing

Spectral Audio Signal Processing

Fourier Theorems for the DTFT

Power Theorem

Repeat (Scaling) Operator

Stretch Operator

We define the stretch operator in the time domain by

$\displaystyle \hbox{\sc Stretch}_{L,n}(x) \isdefs \left\{\begin{array}{ll} x\left(\frac{n}{L}\right), & n = 0\;(\hbox{\sc mod}\ L) \\ [5pt] 0, & \hbox{otherwise} \\ \end{array} \right..$

(3.29)

In other terms, we stretch a sampled signal by the factor

by inserting

zeros in between each pair of samples of the signal.

**Figure 2.1:** Illustration of the stretch operator.
$\includegraphics[width=4in]{eps/stretch2}$

In the literature on multirate filter banks (see Chapter 11), the stretch operator is typically called instead the upsampling operator. That is, stretching a signal by the factor of is called upsampling the signal by the factor . (See §11.1.1 for the graphical symbol ( $\uparrow K$ ) and associated discussion.) The term ``stretch'' is preferred in this book because ``upsampling'' is easily confused with ``increasing the sampling rate''; resampling a signal to a higher sampling rate is conceptually implemented by a stretch operation followed by an ideal lowpass filter which moves the inserted zeros to their properly interpolated values.

Note that we could also call the stretch operator the scaling operator, to unify the terminology in the discrete-time case with that of the continuous-time case (§2.4.1 below).

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``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2022-02-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

Stretch Operator

``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1. Copyright © 2022-02-28 by Julius O. Smith III Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2022-02-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University