CHAPTER 3 Discrete-Time Signals in the Transform-Domain

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CHAPTER 3 Discrete-Time Signals in the
Transform-Domain

• Wang Weilian
• wlwang_at_ynu.edu.cn
• School of Information Science and Technology
• Yunnan University

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Outline

• The Discrete-Time Fourier Transform
• The Discrete Fourier Transform
• Relation between the DTFT and the DFT, and
  
• Their Inverses
• Discrete Fourier Transform Properties
• Computation of the DFT of Real Sequences
• Linear Convolution Using the DFT
• The z-Transform

3
Outline

• Region of Convergence of a Rational z-Transform
• Inverse z-Transform
• z-Transform Properties

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The Discrete-Time Fourier Transform

• The discrete-time Fourier transform (DTFT) or,
  simply, the Fourier transform of a discretetime
  sequence xn is a representation of the sequence
  in terms of the complex exponential sequence
  where is the real frequency variable.
• The discrete-time Fourier transform
  of a sequence xn is defined by

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The Discrete-Time Fourier Transform

• In general is a complex function of
  the real variable and can be written in
  rectangular form as
• where and are,
  respectively, the real and imaginary parts of
  , and are real functions of .
• Polar form

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The Discrete-Time Fourier Transform

• Convergence Condition
• If xn is an absolutely summable sequence,
  i.e.,
• Thus the equation is a sufficient condition
  for the existence of the DTFT.

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The Discrete-Time Fourier Transform

• Bandlimited Signals
• A full-band discrete-time signal has a spectrum
  occupying the whole frequency rang
  .
• If the spectrum is limited to a portion of the
  frequency range , it is called
  a bandlimited signal.
• A lowpass discrete-time signal has a spectrum
  occupying the frequency range
  , where is called the bandwidth of the
  signal.
• A bandpass discrete-time signal has a spectrum
  occupying the frequency range
  , where
  is its bandwidth.

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The Discrete-Time Fourier Transform

• Discrete-Time Fourier Transform Properties
• There are a number of important properties of
  the discrete-time Fourier transform which are
  useful in digital signal processing applications.
  We list the general properties in Table 3.2, and
  the symmetry properties in Tables 3.3 and 3.4.

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The Discrete-Time Fourier Transform

• Energy Density Spectrum

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The Discrete Fourier Transform

• DTFT Computation Using MATLAB
• The Signal Processing Toolbox in MATLAB
• Functions
• freqz
• abs
• Angle
• The forms of freqz
• H freqz(num, den, w)
• H, w freqz(num, den, k, whole)
• Example 3.8 Program 3_1

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The Discrete Fourier Transform

• Definition
• The simplest relation between a
  finite-length sequence xn, defined for
  , and its
• DTFT is obtained by uniformly
  sampling
• on the -axis between
  at
• ,
  .

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The Discrete Fourier Transform

• The sequence Xk is called the discrete Fourier
  transform (DFT) of the sequence xn.
• Using the commonly used notation
• We can rewrite as
• Inverse discrete Fourier transform (IDFT)

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The Discrete Fourier Transform

• Matrix Relations
• The DFT samples defined in
  can
• be expressed in matrix form as
• where X is the vector composed of the N DFT
  samples,
• x is the vector of N input samples,

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The Discrete Fourier Transform

• is the DFT matrix given by
• IDFT relations

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The Discrete Fourier Transform

• DFT computation Using MATLAB
• MATLAB functions
• fft(x), fft(x,N), ifft(X), ifft(X,N)
• X fft(x, N)
• If N lt Rlength(x), truncate (??) to the first
  N samples.
• If N gt Rlength(x), zero-padded (??) at the
  end.
• Example 3.11, 3.12, 3.13, Program 3_2, 3_3, 3_4.

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Relation between the DTFT and the DFT, and their
Inverses

• DTFT from DFT by Interpolation
• We could express in terms of
  Xk

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Relation between the DTFT and the DFT, and their
Inverses

• Sampling the DTFT
• Consider the following question
• We obtain the relation
• Example 3.14

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Relation between the DTFT and the DFT, and their
Inverses

• Numerical Computation of the DTFT Using the DFT
• Let be the DTFT of length-N sequence
  xn. We wish to evaluate at a dense
  grid of frequencies

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Discrete Fourier Transform Properties

• Discrete Fourier Transform Properties
• Like the DTFT, the DFT also satisfies a
  number of properties that are useful in signal
  processing application. A summary of the DFT
  properties are included in Tables 3.5, 3.6, and
  3.7.

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Discrete Fourier Transform Properties

• Circular Shift of a Sequence
• Time-shifting property of the DTFT
• Circular shifting property of the DFT

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Computation of the DFT of Real Sequences

• Computation of the DFT of Real Sequences
• Tow N-point DFTs can be computed efficiently
  using a single N-point DFT Xk of a complex
  length-N sequence xn defined by
• where, and

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Computation of the DFT of Real Sequences

• we arrive at
• Note that

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Linear Convolution Using the DFT

• Linear Convolution of Two Finite-Length Sequences
• Let gn and hn be finite-length sequences
  of lengths N and M, respectively. Denote LMN-1.
  Define two length-L sequences,

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Linear Convolution Using the DFT

• obtained by appending gn and hn with
• zero-valued samples. Then
• Linear Convolution of a Finite-Length Sequence
  with an Infinite-Length Sequence
• Overlap-Add Method
• Overlap-Save Method

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The z-Transform

• Definition
• For a given sequence gn, its z-transform
  G(z) is defined as
• where is a
  complex variable.
• If we let , then the right-hand
  side of the above expression reduces to

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The z-Transform

• For a given sequence, the set R of values of
  z
• for which its z-transform converges is called
  
• the region of convergence (ROC).
• If
• In general, the region of convergence R of a
  z-transform of a sequence gn is an annular
  region of the z-plane

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The z-Transform

• Rational z-Transforms
• An alternate representation as a ration of two
  polynomials in z
• An alternate representation in factored form as

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Region of Convergence of a Rational z-Transform

• The ROC of a rational z-transform is bounded by
  the locations of its poles.
• A finite-length sequence ROC
• A right-sided sequence ROC
• A left-sided sequence ROC
• A two-sided sequence ROC

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Inverse z-Transform

• General Expression
• By the inverse Fourier transform relation. We
  have
• By making the change of variable
  , the above equation can be converted into a
  contour integral given by
• Where is a counterclockwise
  contour of integration defined by

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Inverse z-Transform

• Inverse Transform by Partial-Fraction Expansion
• can be expressed as
• We can divide P(Z) by D(Z) and re-express G(Z) as

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Inverse z-Transform

• Simple Poles p168
• Multiple Poles p169

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z-Transform Properties

• P174 Table 3.9

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Summary

• Three different frequency-domain representations
  of an aperiodic discrete-time sequence have been
  introduced and their properties reviewed .Two of
  these representations, the discrete-time Fourier
  transform (DTFT) and the z-transform, are
  applicable to any arbitrary sequence, whereas the
  third one , the discrete Fourier transform (DFT),
  can be applied only to finite-length sequences.
• Relation between these three transforms have been
  established. The chapter ends with a discussion
  on the transform-domain representation of a
  random discrete-time sequence.
• For future convenience we summarize below these
  three frequency-domain representations.

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Assignment and Experiment

• Assignment
• A03 3.2, 3.12, 3.20, See p180182
• A04
• A05
• Experiment
• E03 Q3.3 See p32
• E04
• E05