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CHAPTER 3 Discrete-Time Signals in the Transform-Domain

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

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CHAPTER 3 Discrete-Time Signals in the Transform-Domain

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  1. CHAPTER 3 Discrete-Time Signals in the Transform-Domain Wang Weilian wlwang@ynu.edu.cn 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 • 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 • 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 • Assignment • 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

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