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1、3.1 The Continuous Time Fourier Translation )3 Discrete-Time Signals in the Frequency Domain3.1.1 Definition of CTFTNormally, is a complex function.(3.1)(3.2)1/163 Discrete-Time Signals in the Frequency DomainIn polar form:In rectangular form:-magnitude spectrum-phase spectrum2/16(a) Finite number o
2、f finite discontinities and a finite number of maxima and minima in any finite interval.(b) Absolutely integrable.Dirichlet conditions: exists if a milder condition of (3.3)(3.3)(3.6)3 Discrete-Time Signals in the Frequency Domain3/161/|Xa(j )| nExample3.1xa(t)t13 Discrete-Time Signals in the Freque
3、ncy Domain4/16nUnit impulse (t)3 Discrete-Time Signals in the Frequency Domain(t)0tt(j )10 The sampling property of the delta functionExample 3.25/16The shifted impulse (t-t0) The sampling property of the delta functionExample 3.3xa(t) =(t-t0)0tt03 Discrete-Time Signals in the Frequency Domain6/16nT
4、he total energy of xa(t) By using (3.2) and conjungating operation3.1.2 Energy Density SpectrumInterchanging the order of integrations 3 Discrete-Time Signals in the Frequency Domain7/16Parsevals Theorem (3.9)Read example 3.4 By yourself. 3 Discrete-Time Signals in the Frequency Domain8/16Energy Den
5、sity Spectrum The energy over a specified range of frequency3 Discrete-Time Signals in the Frequency Domain9/163.1.3 Band-Limited Continuous-Time SignalsnAn ideal band-limited signal has a spectrum that is zero outside a frequency range 0 a| | b, that is3 Discrete-Time Signals in the Frequency Domai
6、n10/16nBand-limited signals are classified according to the frequency range where most of the signals energy is concentrated.nA lowpass continuous-time signal has a spectrum occupying the frequency range 0| |p ,where p is called the bandwidth of the signal.3 Discrete-Time Signals in the Frequency Do
7、main11/16nA highpass contnuous-time signal has a spectrum occupying the frequency range 0 p| | and has a bandwidth from p to . .nA bandpass contnuous-time signal has a spectrum occupying the frequency range 0 L| | H and has a bandwidth of H- L3 Discrete-Time Signals in the Frequency Domain12/16nHowe
8、ver, an ideal band-limited signal can not be generated in practice, and for practical purposes, it is sufficient to ensure that the signal energy is sufficiently small outside the specified frequency range.nA precise definition of the bandwidth depends on applications.3 Discrete-Time Signals in the
9、Frequency Domain13/16nAs can be seen from Figure3.2(a),the signal in example 3.1 is a lowpass signal. It can be shown that 80% energy of this signal is contained in the frequency range 0| |0.4898nHence, we can define the 80% bandwidth to be 0.4898 3 Discrete-Time Signals in the Frequency Domain14/16
10、Type of Property Signal CTFT Differentiation dxa(t)/dt j Xa(j )Frequency-shifting ej 0txa(t) Xa(j( - 0)Time-shifting xa(t-t0) e-j t0Xa(j )Linearity axa(t)+bha(t) aXa(j )+bHa(j )ha(t) Ha(j )xa(t) Xa(j )3.1.4 Properties of CTFT (Appended)3 Discrete-Time Signals in the Frequency Domain15/16Type of Property Sequence CTFT Modulation xa(t)ha(t)Convolution xa(t)* ha(t) Xa(j ) Ha(j )Conjugation x*a(t) X*a(-j ) Conjugate Symmetry real xa(t)Xa(-j )=X*a(j )Parsevals Theorem 3 Discrete-Time Signals in the Frequency Domain16/16
