1 audio signal processing

Amity School of Engineering & Technology

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AUDIO SIGNAL PROCESSING

Credit Units: 4

Mukesh Bhardwaj


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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http://www.safaribooksonline.com/library/view/audio-signal-processing/9780471791478/ch002-sec005.html



• 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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• 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 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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