Xử lý tín hiệu số 6 ZTransforms Biến đổi Z

DSP Dr. Dung Trung Vo 1 DSPChapter 5Dr. Dung Trung Vo Digital Signal Processing zTransforms Dr. Dung Trung Vo Telecommunication Divisions Department of Electrical and Electronics September, 2013 DSPChapter 5Dr. Dung Trung Vo Basic Properties of zTransforms  Usage: ztransforms is used a tool for the analysis, design, and implementation of digital filters.  ztransform: Given a discretetime signal x(n), its ztransform is defined as the following series or  Transfer function: is the ztransform of h(n) CuuDuongThanCong.com https:fb.comtailieudientucnttDSP Dr. Dung Trung Vo 2 DSPChapter 5Dr. Dung Trung Vo Basic Properties of zTransforms  Example 5.1.1: Determine the transfer function H(z) of the two causal filters DSPChapter 5Dr. Dung Trung Vo Basic Properties of zTransforms The three most important properties of ztransforms that facilitate the analysis and synthesis of linear systems are  Linearity property: a linear combination of signals is equal to the linear combination of ztransforms  Delay property: the effect of delaying a signal by D sampling units is equivalent to multiplying its ztransform by a factor zD  Convolution property: convolution in the time domain becomes multiplication in the zdomain CuuDuongThanCong.com https:fb.comtailieudientucnttDSP Dr. Dung Trung Vo 3 DSPChapter 5Dr. Dung Trung Vo Basic Properties of zTransforms  Example: Using the unitstep identity u(n)−u(n − 1)= δ(n), valid for all n, and the ztransform properties, determine the ztransforms of the two signals: DSPChapter 5Dr. Dung Trung Vo Basic Properties of zTransforms  Example: Compute the output of convolution by carrying out the convolution operation as multiplication in the zdomain: CuuDuongThanCong.com https:fb.comtailieudientucnttDSP Dr. Dung Trung Vo 4 DSPChapter 5Dr. Dung Trung Vo Region of Convergence  Region of Convergence: region of convergence (ROC) of the ztransform X(z) is defined to be that subset of the complex zplane C for X(z) converges:  ROC important concept: It allows the unique inversion of the ztransform and provides convenient characterizations of the causality and stability properties of a signal or system. ROC depends on the signal x(n) being transformed DSPChapter 5Dr. Dung Trung Vo Region of Convergence  Example: find ROC of the causal signal: CuuDuongThanCong.com https:fb.comtailieudientucnttDSP Dr. Dung Trung Vo 5 DSPChapter 5Dr. Dung Trung Vo Region of Convergence  Example: find ROC of the anticausal signal: DSPChapter 5Dr. Dung Trung Vo Region of Convergence  General ROC of same ztransform: The two signals have the same ztransform but completely disjoint ROCs where a is any complex number. Their ROCs are shown below CuuDuongThanCong.com https:fb.comtailieudientucnttDSP Dr. Dung Trung Vo 6 DSPChapter 5Dr. Dung Trung Vo Region of Convergence  Example: Determine the ztransform and corresponding region of convergence of the following signals DSPChapter 5Dr. Dung Trung Vo Region of Convergence  Example: Determine the ztransform and corresponding region of convergence of the following signals CuuDuongThanCong.com https:fb.comtailieudientucnttDSP Dr. Dung Trung Vo 7 DSPChapter 5Dr. Dung Trung Vo Region of Convergence  Example: Determine the ztransform and corresponding region of convergence of the following signals DSPChapter 5Dr. Dung Trung Vo Region of Convergence  Example: Determine the ztransform and corresponding region of convergence of the following signals CuuDuongThanCong.com https:fb.comtailieudientucnttDSP Dr. Dung Trung Vo 8 DSPChapter 5Dr. Dung Trung Vo Region of Convergence  Example: Determine the ztransform and corresponding region of convergence of the following signals DSPChapter 5Dr. Dung Trung Vo Region of Convergence Solution: three ROCs are shown below CuuDuongThanCong.com https:fb.comtailieudientucnttDSP Dr. Dung Trung Vo 9 DSPChapter 5Dr. Dung Trung Vo Causality and Stability  Causal signal:  ztransform:  ROC: outside of the circle defined by the pole of maximum magnitude DSPChapter 5Dr. Dung Trung Vo Causality and Stability  Anticausal signal:  ztransform:  ROC: inside of the circle defined by the pole of minimum magnitude CuuDuongThanCong.com https:fb.comtailieudientucnttDSP Dr. Dung Trung Vo 10 DSPChapter 5Dr. Dung Trung Vo Causality and Stability  Causal signal: causal signals are characterized by ROCs that are outside the maximum pole circle  Anticausal signal: Anticausal signals have ROCs that are inside the minimum pole circle  Mixed signal: Mixed signals have ROCs that are the annular region between two circles—with the poles that lie inside the inner circle contributing causally and the poles that lie outside the outer circle contributing anticausally DSPChapter 5Dr. Dung Trung Vo Causality and Stability  Stability in the zdomain of signal x(n): It can be shown that a necessary and sufficient condition for the stability of a signal x(n) is that the ROC of the corresponding ztransform contain the unit circle.  Stability in the zdomain of impulse response h(n): same as h(n).  Characteristics: Stability is not necessarily compatible with causality  Stability for causal signalsystem: all its poles lie strictly inside the unit circle in the zplane  Stability for causal signalsystem: all its poles must lie strictly outside the unit circle CuuDuongThanCong.com https:fb.comtailieudientucnttDSP Dr. Dung Trung Vo 11 DSPChapter 5Dr. Dung Trung Vo Causality and Stability  Stable ROCs:  |pi| < 1, i = 1, 2, 3, 4:  |pi| > 1, i = 1, 2, 3, 4:  |p1| < |p2| < 1 and |p4| > |p3| > 1: DSPChapter 5Dr. Dung Trung Vo Frequency Spectrum  Discretetime Fourier transform (DTFT): or frequency spectrum, frequency content of a signal x(n) is defined  DTFT and ztransform: DTFT is the evaluation of the ztransform on unit circle  Inverse DTFT and ztransform CuuDuongThanCong.com https:fb.comtailieudientucnttDSP Dr. Dung Trung Vo 12 DSPChapter 5Dr. Dung Trung Vo Frequency Spectrum  Frequency response: H(ω) of a linear system h(n) with transfer function H(z) is defined in the same way  DTFT and ztransform: DSPChapter 5Dr. Dung Trung Vo Frequency Spectrum  Digital frequency: ω is in units of radianssample and is related to the physical frequency f in Hz  Nyquist interval:  Fourier spectrum:  Periodic: In units of ω, periodicity in f with period fs becomes periodicity in ω with period 2π. Therefore, X(ω) may be considered only over one period, such as the Nyquist interval  I DTFT: In terms of the physical frequency: CuuDuongThanCong.com https:fb.comtailieudientucnttDSP Dr. Dung Trung Vo 13 DSPChapter 5Dr. Dung Trung Vo Frequency Spectrum  Example: consider a (doublesided) complex sinusoid of frequency ω0  Its DTFT:  Inverse DTFT:  DTFT of realvalued cosine and sine signals: DSPChapter 5Dr. Dung Trung Vo Frequency Spectrum  Parseval’s equation: relation between the total energy of a sequence to its spectrum  Evaluation of ztransform on the unit circle: In order for the spectrum X(ω) to exist, the ROC of the ztransform X(z) must contain the unit circle; otherwise the ztransform will diverge at the unit circle points z = ejω . But if the ROC contains the unit circle, the signal x(n) must be stable. Thus, the Fourier transform X(ω) exists only for stable signals. CuuDuongThanCong.com https:fb.comtailieudientucnttDSP Dr. Dung Trung Vo 14 DSPChapter 5Dr. Dung Trung Vo Frequency Spectrum  Low, medium, and highfrequency parts of the unit circle: DSPChapter 5Dr. Dung Trung Vo Frequency Spectrum  Hermitian property: For realvalued signals x(n) Relationships for the magnitude and phase responses  Filtering in frequency domain: CuuDuongThanCong.com https:fb.comtailieudientucnttDSP Dr. Dung Trung Vo 15 DSPChapter 5Dr. Dung Trung Vo Inverse zTransforms  Task: The problem of inverting a given ztransform X(z) is to find the time signal x(n) whose ztransform is X(z)  Unique: can be made unique by specifying the corresponding ROC  Method to invert a ztransform: convenient to break X(z) into its partial fraction (PF) expansion form, into a sum of individual pole terms of the type where for i = 1, 2, . . . , M DSPChapter 5Dr. Dung Trung Vo Inverse zTransforms  Example: find the PF expansion of equation CuuDuongThanCong.com https:fb.comtailieudientucnttDSP Dr. Dung Trung Vo 16 DSPChapter 5Dr. Dung Trung Vo Inverse zTransforms  Degree of N(z) is equal to the degree of D(z): the PF expansion must be modified by adding an extra term of the form where  Degree of N(z) is greater than the degree of D(z): divide the polynomial D(z) into N(z) DSPChapter 5Dr. Dung Trung Vo Inverse zTransforms  Example: Compute all possible inverse ztransforms of CuuDuongThanCong.com https:fb.comtailieudientucnttDSP Dr. Dung Trung Vo 17 DSPChapter 5Dr. Dung Trung Vo Inverse zTransforms  Example: Compute all possible inverse ztransforms of DSPChapter 5Dr. Dung Trung Vo Inverse zTransforms  Complexvalued poles: because N(z) and D(z) have realvalued coefficients, the complexvalued poles of X(z) come in complexconjugate pairs  Corresponding inverse ztransform: We have Assume Then Final result CuuDuongThanCong.com https:fb.comtailieudientucnttDSP Dr. Dung Trung Vo 18 DSPChapter 5Dr. Dung Trung Vo Inverse zTransforms  Example: Determine all possible inverse ztransforms of DSPChapter 5Dr. Dung Trung Vo Inverse zTransforms  Homework: provided in class CuuDuongThanCong.com https:fb.comtailieudientucntt

Digital Signal Processing z-Transforms Dr Dung Trung Vo Telecommunication Divisions Department of Electrical and Electronics September, 2013 DSP-Chapter 5-Dr Dung Trung Vo Basic Properties of z-Transforms  Usage: z-transforms is used a tool for the analysis, design, and implementation of digital filters  z-transform: Given a discrete-time signal x(n), its z-transform is defined as the following series or  Transfer function: is the z-transform of h(n) DSP-Chapter 5-Dr Dung Trung Vo CuuDuongThanCong.com DSP - Dr Dung Trung Vo https://fb.com/tailieudientucntt Basic Properties of z-Transforms  Example 5.1.1: Determine the transfer function H(z) of the two causal filters DSP-Chapter 5-Dr Dung Trung Vo Basic Properties of z-Transforms The three most important properties of z-transforms that facilitate the analysis and synthesis of linear systems are  Linearity property: a linear combination of signals is equal to the linear combination of z-transforms  Delay property: the effect of delaying a signal by D sampling units is equivalent to multiplying its z-transform by a factor z-D  Convolution property: convolution in the time domain becomes multiplication in the z-domain DSP-Chapter 5-Dr Dung Trung Vo CuuDuongThanCong.com DSP - Dr Dung Trung Vo https://fb.com/tailieudientucntt Basic Properties of z-Transforms  Example: Using the unit-step identity u(n)−u(n − 1)= δ(n), valid for all n, and the z-transform properties, determine the z-transforms of the two signals: DSP-Chapter 5-Dr Dung Trung Vo Basic Properties of z-Transforms  Example: Compute the output of convolution by carrying out the convolution operation as multiplication in the z-domain: DSP-Chapter 5-Dr Dung Trung Vo CuuDuongThanCong.com DSP - Dr Dung Trung Vo https://fb.com/tailieudientucntt Region of Convergence  Region of Convergence: region of convergence (ROC) of the z-transform X(z) is defined to be that subset of the complex z-plane C for X(z) converges:  ROC important concept: It allows the unique inversion of the z-transform and provides convenient characterizations of the causality and stability properties of a signal or system ROC depends on the signal x(n) being transformed DSP-Chapter 5-Dr Dung Trung Vo Region of Convergence  Example: find ROC of the causal signal: DSP-Chapter 5-Dr Dung Trung Vo CuuDuongThanCong.com DSP - Dr Dung Trung Vo https://fb.com/tailieudientucntt Region of Convergence  Example: find ROC of the anticausal signal: DSP-Chapter 5-Dr Dung Trung Vo Region of Convergence  General ROC of same z-transform: The two signals have the same ztransform but completely disjoint ROCs where a is any complex number Their ROCs are shown below DSP-Chapter 5-Dr Dung Trung Vo CuuDuongThanCong.com DSP - Dr Dung Trung Vo https://fb.com/tailieudientucntt Region of Convergence  Example: Determine the z-transform and corresponding region of convergence of the following signals DSP-Chapter 5-Dr Dung Trung Vo Region of Convergence  Example: Determine the z-transform and corresponding region of convergence of the following signals DSP-Chapter 5-Dr Dung Trung Vo CuuDuongThanCong.com DSP - Dr Dung Trung Vo https://fb.com/tailieudientucntt Region of Convergence  Example: Determine the z-transform and corresponding region of convergence of the following signals DSP-Chapter 5-Dr Dung Trung Vo Region of Convergence  Example: Determine the z-transform and corresponding region of convergence of the following signals DSP-Chapter 5-Dr Dung Trung Vo CuuDuongThanCong.com DSP - Dr Dung Trung Vo https://fb.com/tailieudientucntt Region of Convergence  Example: Determine the z-transform and corresponding region of convergence of the following signals DSP-Chapter 5-Dr Dung Trung Vo Region of Convergence Solution: three ROCs are shown below DSP-Chapter 5-Dr Dung Trung Vo CuuDuongThanCong.com DSP - Dr Dung Trung Vo https://fb.com/tailieudientucntt Causality and Stability  Causal signal:  z-transform:  ROC: outside of the circle defined by the pole of maximum magnitude DSP-Chapter 5-Dr Dung Trung Vo Causality and Stability  Anticausal signal:  z-transform:  ROC: inside of the circle defined by the pole of minimum magnitude DSP-Chapter 5-Dr Dung Trung Vo CuuDuongThanCong.com DSP - Dr Dung Trung Vo https://fb.com/tailieudientucntt Causality and Stability  Causal signal: causal signals are characterized by ROCs that are outside the maximum pole circle  Anticausal signal: Anticausal signals have ROCs that are inside the minimum pole circle  Mixed signal: Mixed signals have ROCs that are the annular region between two circles—with the poles that lie inside the inner circle contributing causally and the poles that lie outside the outer circle contributing anticausally DSP-Chapter 5-Dr Dung Trung Vo Causality and Stability  Stability in the z-domain of signal x(n): It can be shown that a necessary and sufficient condition for the stability of a signal x(n) is that the ROC of the corresponding z-transform contain the unit circle  Stability in the z-domain of impulse response h(n): same as h(n)  Characteristics: Stability is not necessarily compatible with causality  Stability for causal signal/system: all its poles lie strictly inside the unit circle in the z-plane  Stability for causal signal/system: all its poles must lie strictly outside the unit circle DSP-Chapter 5-Dr Dung Trung Vo CuuDuongThanCong.com DSP - Dr Dung Trung Vo https://fb.com/tailieudientucntt 10 Causality and Stability  Stable ROCs:  |pi| < 1, i = 1, 2, 3, 4:  |pi| > 1, i = 1, 2, 3, 4:  |p1| < |p2| < and |p4| > |p3| > 1: DSP-Chapter 5-Dr Dung Trung Vo Frequency Spectrum  Discrete-time Fourier transform (DTFT): or frequency spectrum, frequency content of a signal x(n) is defined  DTFT and z-transform: DTFT is the evaluation of the z-transform on unit circle  Inverse DTFT and z-transform DSP-Chapter 5-Dr Dung Trung Vo CuuDuongThanCong.com DSP - Dr Dung Trung Vo https://fb.com/tailieudientucntt 11 Frequency Spectrum  Frequency response: H(ω) of a linear system h(n) with transfer function H(z) is defined in the same way  DTFT and z-transform: DSP-Chapter 5-Dr Dung Trung Vo Frequency Spectrum  Digital frequency: ω is in units of [radians/sample] and is related to the physical frequency f in Hz  Nyquist interval:  Fourier spectrum:  Periodic: In units of ω, periodicity in f with period fs becomes periodicity in ω with period 2π Therefore, X(ω) may be considered only over one period, such as the Nyquist interval  I DTFT: In terms of the physical frequency: DSP-Chapter 5-Dr Dung Trung Vo CuuDuongThanCong.com DSP - Dr Dung Trung Vo https://fb.com/tailieudientucntt 12 Frequency Spectrum  Example: consider a (double-sided) complex sinusoid of frequency ω0  Its DTFT:  Inverse DTFT:  DTFT of real-valued cosine and sine signals: DSP-Chapter 5-Dr Dung Trung Vo Frequency Spectrum  Parseval’s equation: relation between the total energy of a sequence to its spectrum  Evaluation of z-transform on the unit circle: In order for the spectrum X(ω) to exist, the ROC of the z-transform X(z) must contain the unit circle; otherwise the z-transform will diverge at the unit circle points z = ejω But if the ROC contains the unit circle, the signal x(n) must be stable Thus, the Fourier transform X(ω) exists only for stable signals DSP-Chapter 5-Dr Dung Trung Vo CuuDuongThanCong.com DSP - Dr Dung Trung Vo https://fb.com/tailieudientucntt 13 Frequency Spectrum  Low-, medium-, and high-frequency parts of the unit circle: DSP-Chapter 5-Dr Dung Trung Vo Frequency Spectrum  Hermitian property: For real-valued signals x(n) Relationships for the magnitude and phase responses  Filtering in frequency domain: DSP-Chapter 5-Dr Dung Trung Vo CuuDuongThanCong.com DSP - Dr Dung Trung Vo https://fb.com/tailieudientucntt 14 Inverse z-Transforms  Task: The problem of inverting a given z-transform X(z) is to find the time signal x(n) whose z-transform is X(z)  Unique: can be made unique by specifying the corresponding ROC  Method to invert a z-transform: convenient to break X(z) into its partial fraction (PF) expansion form, into a sum of individual pole terms of the type where for i = 1, 2, , M DSP-Chapter 5-Dr Dung Trung Vo Inverse z-Transforms  Example: find the PF expansion of equation DSP-Chapter 5-Dr Dung Trung Vo CuuDuongThanCong.com DSP - Dr Dung Trung Vo https://fb.com/tailieudientucntt 15 Inverse z-Transforms  Degree of N(z) is equal to the degree of D(z): the PF expansion must be modified by adding an extra term of the form where  Degree of N(z) is greater than the degree of D(z): divide the polynomial D(z) into N(z) DSP-Chapter 5-Dr Dung Trung Vo Inverse z-Transforms  Example: Compute all possible inverse z-transforms of DSP-Chapter 5-Dr Dung Trung Vo CuuDuongThanCong.com DSP - Dr Dung Trung Vo https://fb.com/tailieudientucntt 16 Inverse z-Transforms  Example: Compute all possible inverse z-transforms of DSP-Chapter 5-Dr Dung Trung Vo Inverse z-Transforms  Complex-valued poles: because N(z) and D(z) have real-valued coefficients, the complex-valued poles of X(z) come in complex-conjugate pairs  Corresponding inverse z-transform: We have Assume Then Final result DSP-Chapter 5-Dr Dung Trung Vo CuuDuongThanCong.com DSP - Dr Dung Trung Vo https://fb.com/tailieudientucntt 17 Inverse z-Transforms  Example: Determine all possible inverse z-transforms of DSP-Chapter 5-Dr Dung Trung Vo Inverse z-Transforms  Homework: provided in class DSP-Chapter 5-Dr Dung Trung Vo CuuDuongThanCong.com DSP - Dr Dung Trung Vo https://fb.com/tailieudientucntt 18 ... D(z) into N(z) DSP-Chapter 5-Dr Dung Trung Vo Inverse z-Transforms  Example: Compute all possible inverse z-transforms of DSP-Chapter 5-Dr Dung Trung Vo CuuDuongThanCong.com DSP - Dr Dung Trung... Inverse z-Transforms  Example: Compute all possible inverse z-transforms of DSP-Chapter 5-Dr Dung Trung Vo Inverse z-Transforms  Complex-valued poles: because N(z) and D(z) have real-valued... https://fb.com/tailieudientucntt 17 Inverse z-Transforms  Example: Determine all possible inverse z-transforms of DSP-Chapter 5-Dr Dung Trung Vo Inverse z-Transforms  Homework: provided in class DSP-Chapter 5-Dr Dung Trung