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1、Introductionn In time-domain, the LTI system is characterized by h(t) or hn;n In frequency-domain, the LTI system is characterized by H(j) or H(ej).n In analyzing LTI system, it is often particularly convenient to utilize the frequency domain. differential and difference equations and convolution op
2、erations in the time domain become algebraic operations in the frequency domain.n In system design, there are typically both time-domain and frequency-domain considerations.6.1 The Magnitude-phase Representation of the Fourier TransformFor signal x(t) :)()()()(jXjFTejXjXtx)()()(jeXjjjDFTeeXeXnx For
3、signal xn :)()()()(jjeXjXeXjX、 Magnitude spectrum Phase spectrum )2cos(32)2cos()2cos(211)(321ttttx032112, 8, 432193. 0, 7 .2, 63212 . 7, 1 . 4, 2 . 1321),(21jjP| ),(|21jjP12:|(,)|: 0MagnitudeP jjPhase12:1:(,)MagnitudePhaseP jj12:(,)MagnitudePhaseP jjConclusion: Effects of Phase Not on signal energy
4、distribution as a function of frequency Can have dramatic effect on signal shape/characterConstructive/Destructive interference Is that important? Depends on the signal and the context6.2 The Magnitude-phase Representation of the Frequency Response of LTI SystemContinuous-time System characterizatio
5、n:Impulse response:Frequency response:)()(jHthF)()()(jXjYjH)()()()()()(jHjXjYjHjXjYgainPhase shiftDiscrete-time System characterization:Impulse response:Frequency response:)()()()()()(jjjjjjeHeXeYeHeXeYgainPhase shift)(jFeHnh)()()(jjjeXeYeH6.2.1 Linear and Nonlinear PhaseLinear phase:Nonlinear phase
6、:Example:Result: Linear phase simply a rigid shift in time, no distortion（Magnitude Response is constant)Nonlinear phase distortion as well as shiftkjH)(functionNonlinearjH)()()()()()(000phaseLineartjHejHttxtytjLinear phaseX2(j) = X1(j)e-j(Linear phase)(Nonlinear phase)(Original signal)Effect of Lin
7、ear and Nonlinear PhaseAll-Pass System1| )(|1| )(|jeHjHq The characteristics of an all-pass system are completely determined by its phase-shift characteristics.6.2.2 Group DelayDefinition:Example:)()(jHdd)()()()()()(0000delaysignalttjHejHttxtytjNote: if the values of are restricted to lie between an
8、d -, we obtain the principal-phase function.)(jHLinear with near 0Impulse response and output of an all-pass system with nonlinear phase6.2.3 Log-Magnitude and Bode PlotsMagnitude spectrum:Phase spectrum:1010|()|20log |()|log()H jH jBod plots10() () log()H jH jBod plots(a logarithmic scale for frequ
9、ency in CT)For real-valued signals and systems, plot for 0.2、LCCDEA Typical Bode plot for a second-order system20log|H(j)| and H(j) vs. log40dB decadeChanges by -Note: ( in discrete-time system) The magnitudes of Fourier transform and frequency responses are often displayed in dB for the same reason
10、s that they are in continuous time. However, for real hn we need only plot for 0 (with linear scale)Lowpass filter:(1) Continuous time:(2) Discrete time:ccjH| , 0| , 1)(1, |()0,|cjcH e6.3 Time-Domain Properties of Ideal Frequency-selective Filterssin( )cth ttImpulse response of Ideal lowpass filters
11、in cnh nnImpulse response of Ideal Lowpass filterStep response of Ideal Lowpass filtertr=Rise timetdhts)()(1)0()()(jHdhsOvershoot by 9%Ringing (Gibbs henomenon)rippleBasic parameter of lowpass filter:6.4 Time-Domain and Frequency-domain Aspects of Non-ideal Filtersl Sometimes we dont want a sharp cu
12、toff.l Often have specifications in time and frequency domain Trade-offs l Realization: anticausal h(t)Homework: 6.5 6.23 6.27CT Rational Frequency Responses If the system is described by LCCDEs (Linear Constant-Coefficient Differential Equations), then kFkkjdtd)(iiNkkkMkkkjHjajbjH)()()()(00Hi(j) =
13、First or Second-order factors22221)(2)()(11)(nnnjjjHjjH First-order system, has only one energy storing element, e.g. L or C. Second-order system, has two energy storing elements, e.g. L and C.Prototypical SystemsDT Rational Frequency Responses If the system is described by LCCDEs (Linear Constant-C
14、oefficient Difference Equations), then kjjDFTkjjDFTeeXknxeeYkny)(,)( ijiNkkjkMkkjkjeHeaebeH)()()()(00Hi(j) = First or Second-order factors0 , 10,cos211)(1,11)(2221rerereHaaeeHjjjjj First-order system, has only one energy storing element, e.g. L or C. Second-order system, has two energy storing elements, e.g. L and C.Prototypical Systems