測試信號英文版課件：Chapter4 LTI Discrete-Time Systems in the Transform-Domain_Lec1

1、Chapter 4LTI Discrete-Time Systemsin the Transform-Domain1LTI Discrete-Time Systems in the Transform DomainIt is easier to design and implement these systems in the transform-domain for certain applications.Such transform-domain representations provide additional insight into the behavior of such sy

2、stems.In this chapter, we will use the DTFT and the z-transform in developing the transform-domain representations of an LTI system23The Frequency ResponseAn important property of an LTI system is that for certain types of input signals, called eigen functions, the output signal is the input signal

3、multiplied by a complex constantWe consider here one such eigen function as the inputConsider the LTI discrete-time system with an impulse response hn shown belowxnhnyn4Its input-output relationship in the time-domain is given by the convolution sumIf the input is of the formthen it follows that the

4、 output is given byThe Frequency Response5The Frequency ResponseThen we can writeThe quantity is called the frequency response of the LTI discrete-time system provides a frequency-domain description of the system is precisely the DTFT of the impulse response hn of the systemDefinition:The Frequency

5、Response6nLTIhnTime-domainFrequency-domain7The Frequency ResponseLTITime-domainFrequency-domain8The Frequency Response , in general, is a complex function of w with a period 2pIt can be expressed in terms of its real and imaginary partsor, in terms of its magnitude and phase,where9The Frequency Resp

6、onseNote: Magnitude and phase functions are real functions of w, whereas the frequency response is a complex function of wIf the impulse response hn is real then the magnitude function is an even function of w: （refer to Table 3.2: DTFT Properties: Symmetry Relations）and the phase function is an odd

7、 function of w:10The Frequency ResponseExample - Consider the M-point moving average filter with an impulse response given byDTFT of hn is:11Frequency Response Computation Using MATLABthe magnitude and gain responses of an M-point moving average filter are shown below12The Transfer FunctionThe convo

8、lution sum description of an LTI discrete-time system with an impulse response hn is given by = xn hn*13The Transfer FunctionAccording to the properties of Z-transform, we have Y(z) = H(z)X(z) Hence,The function H(z), which is the z-transform of the impulse response hn of the LTI system, is called t

9、he transfer function or the system functionThe inverse z-transform of the transfer function H(z) yields the impulse response hn14The Transfer FunctionOr, equivalently asAn alternate form of the transfer function is given by15The Transfer FunctionOr, equivalently as are the finite zeros, and are the

10、finite poles of H(z)If N M, there are additional zeros at z = 0If N M, there are additional poles at z = 016The Transfer FunctionFor a causal IIR digital filter, the impulse response is a causal sequenceThe ROC of the causal transfer functionis thus exterior to a circle going through the pole farthe

11、st from the originThus the ROC is given by17The Transfer FunctionExample - Consider the M-point moving-average FIR filter with an impulse responseIts transfer function is then given by18The Transfer FunctionThe transfer function has M zeros on the unit circle at ,There are poles at z = 0 and a singl

12、e pole at z = 1The pole at z = 1 exactly cancels the zero at z = 1The ROC is the entire z-plane except z = 0M = 819The Transfer FunctionExample - A causal LTI IIR digital filter is described by a constant coefficient difference equation given byIts transfer function is therefore given by20The Transf

13、er FunctionAlternate forms:Note: Poles farthest from z = 0 have a magnitudeROC: z,p,k=tf2zp(num,den)21Frequency Response from Transfer FunctionIf the ROC of the transfer function H(z) includes the unit circle, then the frequency response of the LTI digital filter can be obtained simply as follows:Fo

14、r a real coefficient transfer function H(z) it can be shown that22Stability Condition in Terms of the Pole LocationsA causal LTI digital filter is BIBO stable if and only if its impulse response hn is absolutely summable, i.e.,We now develop a stability condition in terms of the pole locations of th

15、e transfer function H(z)S23z-TransformFrom our earlier discussion on the uniform convergence of the DTFT, it follows that the seriesconverges if is absolutely summable, i.e., if24Stability Condition in Terms of the Pole LocationsThus, if the ROC includes the unit circle |z| =r= 1, then the digital f

16、ilter is stable, and vice versaAn FIR digital filter with bounded impulse response is always stableROC includes |z|=1 Stable system25Stability Condition in Terms of the Pole LocationsA stable IIR digital filter, its ROC ROC includes |z|=1 Stable system26Stability Condition in Terms of the Pole Locat

17、ionsOn the other hand, an IIR filter may be unstable if not designed properlyIn addition, an originally stable IIR filter characterized by infinite precision coefficients may become unstable when coefficients get quantized due to implementation27Stability Condition in Terms of the Pole LocationsExam

18、ple - Consider the causal IIR transfer functionThe plot of the impulse response coefficients is shown on the next slide28Stability Condition in Terms of the Pole LocationsAs can be seen from the above plot, the impulse response coefficient hn decays rapidly to zero value as n increaseshn29Stability

19、Condition in Terms of the Pole LocationsThe absolute summability condition of hn is satisfiedHence, H(z) is a stable transfer functionNow, consider the case when the transfer function coefficients are rounded to values with 2 digits after the decimal point:30Stability Condition in Terms of the Pole

20、LocationsA plot of the impulse response of is shown below31Stability Condition in Terms of the Pole LocationsIn this case, the impulse response coefficient increases rapidly to a constant value as n increasesHence, the absolute summability condition of is violatedThus, is an unstable transfer functi

21、on32Stability Condition in Terms of the Pole LocationsThe stability testing of a IIR transfer function is therefore an important problemIn most cases it is difficult to compute the infinite sumFor a causal IIR transfer function, an alternate, easy-to-test, stability condition is developed nextS33Sta

22、bility Condition in Terms of the Pole LocationsConsider the causal IIR digital filter with a rational transfer function H(z) given byIts impulse response hn is a right-sided sequenceThe ROC of H(z) is exterior to a circle going through the pole farthest from z = 034Stability Condition in Terms of the Pole LocationsBut stability requires that the ROC of the z-transform H(z) includes the unit circle, then 35Stability Condition in Terms of the Pole LocationsConclusion: All poles of a causal stable transfer function H(z) must be strictly inside the unit circle The stability region