1. CHAPTER 3 Discrete-Time Signals in the Transform-Domain
Wang Weilian
School of Information Science and Technology
Yunnan University

2. 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
云南大学滇池学院课程：数字信号处理

3. Outline Region of Convergence of a Rational z-Transform
Inverse z-Transform
z-Transform Properties
云南大学滇池学院课程：数字信号处理

4. The Discrete-Time Fourier Transform
The discrete-time Fourier transform (DTFT) or, simply, the Fourier transform of a discrete–time sequence x[n] 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 x[n] is defined by
云南大学滇池学院课程：数字信号处理

5. 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
云南大学滇池学院课程：数字信号处理

6. The Discrete-Time Fourier Transform
Convergence Condition:
If x[n] is an absolutely summable sequence, i.e.,
Thus the equation is a sufficient condition for the existence of the DTFT.
云南大学滇池学院课程：数字信号处理

7. 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.
云南大学滇池学院课程：数字信号处理

8. 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.
云南大学滇池学院课程：数字信号处理

9. The Discrete-Time Fourier Transform
Energy Density Spectrum
云南大学滇池学院课程：数字信号处理

10. 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
云南大学滇池学院课程：数字信号处理

11. The Discrete Fourier Transform
Definition
The simplest relation between a finite-length sequence x[n], defined for , and its
DTFT is obtained by uniformly sampling
on the axis between at ,
云南大学滇池学院课程：数字信号处理

12. The Discrete Fourier Transform
The sequence X[k] is called the discrete Fourier transform (DFT) of the sequence x[n].
Using the commonly used notation
We can rewrite as
Inverse discrete Fourier transform (IDFT)
云南大学滇池学院课程：数字信号处理

13. 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,
云南大学滇池学院课程：数字信号处理

14. The Discrete Fourier Transform
is the DFT matrix given by
IDFT relations
云南大学滇池学院课程：数字信号处理

15. 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 < R=length(x), truncate (截短) to the first N samples.
If N > R=length(x), zero-padded (补零) at the end.
Example 3.11, 3.12, 3.13, Program 3_2, 3_3, 3_4.
云南大学滇池学院课程：数字信号处理

16. Relation between the DTFT and the DFT, and their Inverses
DTFT from DFT by Interpolation
We could express in terms of X[k]:
云南大学滇池学院课程：数字信号处理

17. Relation between the DTFT and the DFT, and their Inverses
Sampling the DTFT
Consider the following question
We obtain the relation
Example 3.14
云南大学滇池学院课程：数字信号处理

18. 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 x[n]. We wish to evaluate at a dense grid of frequencies:
云南大学滇池学院课程：数字信号处理

19. 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.
云南大学滇池学院课程：数字信号处理

20. Discrete Fourier Transform Properties
Circular Shift of a Sequence
Time-shifting property of the DTFT
Circular shifting property of the DFT
云南大学滇池学院课程：数字信号处理

21. Computation of the DFT of Real Sequences
Tow N-point DFTs can be computed efficiently using a single N-point DFT X[k] of a complex length-N sequence x[n] defined by
where, and
云南大学滇池学院课程：数字信号处理

22. Computation of the DFT of Real Sequences
we arrive at:
Note that
云南大学滇池学院课程：数字信号处理

23. Linear Convolution Using the DFT
Linear Convolution of Two Finite-Length Sequences
Let g[n] and h[n] be finite-length sequences of lengths N and M, respectively. Denote L=M+N-1. Define two length-L sequences,
云南大学滇池学院课程：数字信号处理

24. Linear Convolution Using the DFT
obtained by appending g[n] and h[n] with
zero-valued samples. Then
Linear Convolution of a Finite-Length Sequence with an Infinite-Length Sequence
Overlap-Add Method
Overlap-Save Method
云南大学滇池学院课程：数字信号处理

25. The z-Transform Definition
For a given sequence g[n], 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
云南大学滇池学院课程：数字信号处理

26. 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 g[n] is an annular region of the z-plane:
云南大学滇池学院课程：数字信号处理

27. The z-Transform Rational z-Transforms
An alternate representation as a ration of two polynomials in z:
An alternate representation in factored form as
云南大学滇池学院课程：数字信号处理

28. 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:
云南大学滇池学院课程：数字信号处理

29. 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
云南大学滇池学院课程：数字信号处理

30. 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
云南大学滇池学院课程：数字信号处理

31. Inverse z-Transform Simple Poles p168 Multiple Poles p169
云南大学滇池学院课程：数字信号处理

32. z-Transform Properties
P174 Table 3.9
云南大学滇池学院课程：数字信号处理

33. 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.
云南大学滇池学院课程：数字信号处理

34. Assignment and Experiment
A03: 3.2, 3.12, 3.20, See p180~182
A04:
A05:
Experiment
E03: Q3.3 See p32
E04:
E05
云南大学滇池学院课程：数字信号处理