Stretch/Repeat (Scaling) Theorem

International Standard Book Number — Click for https://mathworld.wolfram.com/ISBN.html

The Fourier Transform (FT) gives the (complex) spectrum of a continuous-time signal of any length. — Click for https://ccrma.stanford.edu/~jos/mdft/Fourier_Transform_FT_Inverse.html

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

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

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

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

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

dc stands for direct current in electrical engineering and signal processing (as opposed to 'alternating current'). The mean (average value) of a signal may be called its dc component. Similarly, frequency zero is called the dc frequency, because a sinusoid with zero frequency is a constant signal. — Click for https://ccrma.stanford.edu/~jos/filters/Mathematical_Sine_Wave_Analysis.html

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Stretch/Repeat (Scaling) Theorem

$\begin{eqnarray*} \hbox{\sc DTFT}_\omega[\hbox{\sc Stretch}_L(x)] &\isdef & \sum_{n=-\infty}^{\infty}\hbox{\sc Stretch}_{L,n}(x)e^{-j\omega n}\\ &=& \sum_{m=-\infty}^{\infty}x(m)e^{-j\omega m L}\qquad \hbox{($m\isdef n/L$)}\\ &\isdef & X(\omega L) \end{eqnarray*}$

As $\omega$ traverses the interval $[-\pi,\pi)$ , $X(\omega L)$ traverses the unit circle

times, thus implementing the repeat operation on the unit circle. Note also that when $\omega = 0$ , we have $\omega L = 0$ , so that dc always maps to dc. At half the sampling rate $\omega=\pm\pi$ , on the other hand, after the mapping, we may have either $Y(\pi)=X(-\pi)$ (

odd), or

(

even), where $Y(\omega) \isdeftext X(\omega L)$ .

The stretch theorem makes it clear how to do ideal sampling-rate conversion for integer upsampling ratios

: We first stretch the signal by the factor

(introducing

zeros between each pair of samples), followed by an ideal lowpass filter cutting off at $\pi/L$ . That is, the filter has a gain of 1 for $\left\vert\omega\right\vert <\pi/L$ , and a gain of 0 for $\pi/L < \left\vert\omega\right\vert \leq \pi$ . Such a system (if it were realizable) implements ideal bandlimited interpolation of the original signal by the factor

The stretch theorem is analogous to the scaling theorem for continuous Fourier transforms (introduced in §2.4.1 below).

Stretch/Repeat (Scaling) Theorem

``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