This document discusses discrete-time signal processing and audio signal processing. It covers topics like discrete-time signals, the z-transform, discrete Fourier transform (DFT) and fast Fourier transform (FFT). The key points are: - Audio signals are typically sampled at 44.1 kHz and quantized to 16 bits per sample. - The z-transform and discrete Fourier transform (DTFT) are used to analyze discrete-time signals in the transform domain, similar to the Laplace transform and continuous-time Fourier transform for analog signals. - The discrete Fourier transform (DFT) provides a computational tool to calculate Fourier transforms by sampling the frequency domain at discrete points, resulting in periodicity in the time and

1. Amity School of Engineering & Technology
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Amity School of Engineering & Technology
AUDIO SIGNAL PROCESSING
Credit Units: 4
Mukesh Bhardwaj

2. Amity School of Engineering & Technology
Module 1
DISCRETE-TIME SIGNAL
PROCESSING
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DISCRETE-TIME SIGNAL PROCESSING
• Audio coding algorithms operate on a quantized
discrete-time signal.
• Prior to compression, most algorithms require that the
audio signal is acquired with high-fidelity characteristics.
• In typical standardized algorithms, audio is assumed to
be bandlimited at 20 kHz, sampled at 44.1 kHz, and
quantized at 16 bits per sample.
• In the following discussion, we will treat audio as a
sequence, i.e., as a stream of numbers denoted
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Transforms for Discrete-Time Signals
• Discrete-time signals are described in the
transform domain using the z-transform and the
discrete-time Fourier transform (DTFT).
• These two transformations have similar roles as
the Laplace transform and the CFT for analog
signals, respectively.
• The z-transform is defined as
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Transforms for Discrete-Time Signals
• where z is a complex variable. Note that if the z-transform
is evaluated on the unit circle, i.e., for
• then the z-transform becomes the discrete-time Fourier
transform (DTFT). The DTFT is given by,
• The DTFT is discrete in time and continuous in
frequency. As expected, the frequency spectrum
associated with the DTFT is periodic with period 2π rads.
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The Discrete and the Fast Fourier Transform
• A computational tool for Fourier transforms is developed
by starting from the DTFT analysis expression (2.11),
and considering a finite length signal consisting
of N points, i.e.,
• Furthermore, the frequency-domain signal is sampled
uniformly at N points within one period, Ω = 0 to 2π, i.e.,
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The Discrete and the Fast Fourier Transform
• With the sampling in the frequency domain, the Fourier
sum of Eq. (2.13) becomes
• It is typical in the DSP literature to replace Ωk with the
frequency index k and hence Eq. (2.15) can be written
as,
• The expression in (2.16) is called the discrete Fourier
transform (DFT).
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The Discrete and the Fast Fourier Transform
• Note that the sampling in the frequency domain forces
periodicity in the time domain, i.e., x(n) = x(n + N).
• We also have periodicity in the frequency domain, X(k)
= X(k + N), because the signal in the time domain is also
discrete.
• These periodicities create circular effects when
convolution is performed by frequency-domain
multiplication, i.e.,
where
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The Discrete and the Fast Fourier Transform
• The symbol ⊗ stands for circular or periodic convolution;
and mod N implies modulo N subtraction of indices.
• With the proper normalization, the DFT matrix can be
written as a unitary matrix.
• The N-point inverse DFT (IDFT) is written as
• The DFT transform pair is represented by the following
notation:
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• The DFT can be computed efficiently using the fast Fourier
transform (FFT).
• The FFT takes advantage of redundancies in the DFT sum by
decimating the sequence into subsequences with even and odd
indices.
• It can be shown that if N is a radix-2 integer, the N-point DFT can
be computed using a series of butterfly stages.
• The complexity associated with the DFT algorithm is of the order
of N2 computations.
• In contrast, the number of computations associated with the FFT
algorithm is roughly of the order of N log2N.
• This is a significant reduction in computational complexity and FFTs
are almost always used in lieu of a DFT.
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