测试信号英文版课件：Chapter5 Digital Processing of Continuous-Time Signals

1、Chapter 5Digital Processing of Continuous-Time Signals2Digital Processing of Continuous-Time SignalsDigital processing of a continuous-time signal involves the following basic steps:(1) Conversion of the continuous-time signal into a discrete-time signal,(2) Processing of the discrete-time signal,(3

2、) Conversion of the processed discrete-time signal back into a continuous-time signal3Digital Processing of Continuous-Time SignalsConversion of a continuous-time signal into digital form is carried out by an analog-to-digital (A/D) converterThe reverse operation of converting a digital signal into

3、a continuous-time signal is performed by a digital-to-analog (D/A) converter4Digital Processing of Continuous-Time SignalsSince the A/D conversion takes a finite amount of time, a sample-and-hold (S/H) circuit is used to ensure that the analog signal at the input of the A/D converter remains constan

4、t in amplitude until the conversion is complete to minimize the error in its representation5Digital Processing of Continuos-Time SignalsTo prevent aliasing, an analog anti-aliasing filter is employed before the S/H circuit To smooth the output signal of the D/A converter, which has a staircase-like

5、waveform, an analog reconstruction filter is usedBoth the anti-aliasing filter and the reconstruction filter are analog lowpass filters6Digital Processing of Continuous-Time Signals Complete block-diagramAnti-aliasingfilterS/HA/DD/ADigitalprocessorReconstructionfilter7Sampling of Continuous-Time Sig

6、nalsAs indicated earlier, discrete-time signals in many applications are generated by sampling continuous-time signalsWe will see that identical discrete-time signals may result from the sampling of more than one distinct continuous-time function (aliasing)8Sampling of Continuous-Time SignalsIn fact

7、, there exists an infinite number of continuous-time signals, which when sampled lead to the same discrete-time signal910Sampling of Continuous-Time SignalsHowever, under certain conditions, it is possible to relate a unique continuous-time signal to a given discrete-time signalsIf these conditions

8、hold, then it is possible to recover the original continuous-time signal from its sampled valuesWe next develop this correspondence and the associated conditions11Effect of Sampling in the Frequency DomainLet be a continuous-time signal that is sampled uniformly at t = nT, generating the sequence gn

9、 wherewith T being the sampling periodThe reciprocal of T is called the sampling frequency , i.e.,12Effect of Sampling in the Frequency DomainNow, the frequency-domain representation of is given by its continuos-time Fourier transform (CTFT):The frequency-domain representation of gn is given by its

10、discrete-time Fourier transform (DTFT):13Effect of Sampling in the Frequency DomainTo establish the relation between and , we treat the sampling operation mathematically as a multiplication of by a periodic impulse train p(t):14Effect of Sampling in the Frequency Domainp(t) consists of a train of id

11、eal impulses with a period T as shown belowThe multiplication operation yields an impulse train:15Effect of Sampling in the Frequency Domain is a continuous-time signal consisting of a train of uniformly spaced impulses with the impulse at t = nT weighted by the sampled value of at that instant16Eff

12、ect of Sampling in the Frequency DomainIllustration of the frequency-domain effects of time-domain sampling17Effect of Sampling in the Frequency DomainHence, the CTFT of is given byTherefore, is a periodic function of W consisting of a sum of shifted and scaled replicas of , shifted by integer multi

13、ples of and scaled by18Effect of Sampling in the Frequency DomainThe term on the RHS of the previous equation for k = 0 is the baseband portion of , and each of the remaining terms are the frequency translated portions ofThe frequency range is called the baseband or Nyquist band19Effect of Sampling

14、in the Frequency DomainIt is evident from the top figure on the previous slide that if , there is no overlap between the shifted replicas of generatingOn the other hand, as indicated by the figure on the bottom, if , there is an overlap of the spectra of the shifted replicas of generating20Effect of

15、 Sampling in the Frequency DomainThus, if , can be recovered exactly from by passing it through an ideal lowpass filter with a gain T and a cutoff frequency greater than and less than as shown below21Effect of Sampling in the Frequency DomainThe spectra of the filter and pertinent signals are shown

16、below22Effect of Sampling in the Frequency DomainOn the other hand, if , due to the overlap of the shifted replicas of , the spectrum cannot be separated by filtering to recover because of the distortion caused by a part of the replicas immediately outside the baseband folded back or aliased into th

17、e baseband23Effect of Sampling in the Frequency Domain (Shannons) Sampling theorem - Let be a band-limited signal with CTFT for Then is uniquely determined by its samples , ifwhere or24Effect of Sampling in the Frequency DomainThe condition is often referred to as the Nyquist conditionThe frequency

18、is usually referred to as the folding frequencyRelationship between and :Compare Eq. (3.63), Eq. (3.66) and Eq. (3.70)25Effect of Sampling in the Frequency DomainGiven , we can recover exactly by generating an impulse train and then passing it through an ideal lowpass filter with a gain T and a cuto

19、ff frequency satisfying26Effect of Sampling in the Frequency DomainThe highest frequency contained in is usually called the Nyquist frequency since it determines the minimum sampling frequency that must be used to fully recover from its sampled versionThe frequency is called the Nyquist rate27Effect

20、 of Sampling in the Frequency DomainOversampling - The sampling frequency is higher than the Nyquist rateUndersampling - The sampling frequency is lower than the Nyquist rateCritical sampling - The sampling frequency is equal to the Nyquist rateNote: A pure sinusoid may not be recoverable from its c

21、ritically sampled versionSeveral Terms28XImpulse to sequence(a) C/Dsequence to impulse Hr(j) (b) D/CT小结：30Effect of Sampling in the Frequency DomainExample 1In digital telephony, a 3.4 kHz signal bandwidth is acceptable for telephone conversationHere, a sampling rate of 8 kHz, which is greater than

22、twice the signal bandwidth, is used31Effect of Sampling in the Frequency DomainExample 2In high-quality analog music signal processing, a bandwidth of 20 kHz has been determined to preserve the fidelityHence, in compact disc (CD) music systems, a sampling rate of 44.1 kHz, which is slightly higher t

23、han twice the signal bandwidth, is usedEffect of Sampling in the Frequency Domain应用系统信号上限频率采样频率地质勘探500 Hz12 kHz生物医学1 kHz24 kHz机械振动2 kHz410 kHz语音4 kHz8-16 kHz音乐20 kHz4096 kHz视频4 MHz8-10 MHz32典型的数字信号处理应用中使用的采样频率Effect of Sampling in the Frequency DomainExample 3333hz,7hz,13hzEffect of Sampling in the Fre