数字信号处理教学课件：3-1 The Continuous Time Fourier Translation )

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 Finite number of f

2、inite 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)|Example3.1xa(t)t13 Discrete-Time Signals in the Frequency Do

3、main4/16Unit 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 Domain6/16The total

4、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 Density Spec

5、trum The energy over a specified range of frequency3 Discrete-Time Signals in the Frequency Domain9/163.1.3 Band-Limited Continuous-Time SignalsAn ideal band-limited signal has a spectrum that is zero outside a frequency range 0a| | b, that is3 Discrete-Time Signals in the Frequency Domain10/16Band-

6、limited signals are classified according to the frequency range where most of the signals energy is concentrated.A 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 Domain11/16A high

7、pass contnuous-time signal has a spectrum occupying the frequency range 0 p| and has a bandwidth from p to .A bandpass contnuous-time signal has a spectrum occupying the frequency range 0L|H and has a bandwidth of H- L3 Discrete-Time Signals in the Frequency Domain12/16However, an ideal band-limited

8、 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.A precise definition of the bandwidth depends on applications.3 Discrete-Time Signals in the Frequency Domain13/16As can

9、 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|0.4898Hence, we can define the 80% bandwidth to be 0.4898 3 Discrete-Time Signals in the Frequency Domain14/16Type of Property Signal CTFT Di

10、fferentiation dxa(t)/dt jXa(j)Frequency-shifting ej0txa(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)