CHAPTER 3 Discrete-Time Signals in the Transform-Domain Wang Weilian wlwang@ynu.edu.cn School of Information Science and Technology Yunnan University

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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