Chapter 7 Sampling School of Electronics & Information Engineering, South China University of Technology OutlineWhat is the signal sampling?Why sampling?How to sample?Theoretical derivation of sampling theoremSampling TheoremChapter 7 Sampling What is sampling Definition of signal sampling 7. SamplingSamplingSampling, taking snap shots of , taking snap shots of x(t) every T seconds. T is the sampling period. f = 1/T is the sampling frequencyAfter the signal sampling ,we can get the samples xk. Why to sample Motivation to signal sampling 7. SamplingDiscrete-Time SystemA/DD/Ainputx(t)xkykoutputy(t)Discrete-time processing of continuous-time signals(1) Good stability in signal transmission(2) Small occupied space Advantages:(3) Low transmission cost(4) Processing discrete-time signal is more flexible 7. Samplingx,Fs,Bits=wavread(motherland);play(x)Fs=44,100 ; Bits=16Sampling Frequency fs=44,100 HzSampling Frequency fs=5,512 HzHow many samples would be adequate for a specific signal?How to sample?The choice of sampling periodHow to sample?The choice of sampling period 7. SamplingHow do we choose sampling period T?Theoretical derivation of sampling theorem ? 7. SamplingTheoretical derivation of sampling theorem 7. Sampling7.1.1 Impulse-train Samplingx(t)p(t)xp(t)(1) Sampling 7. SamplingTheoretical derivation of sampling theoremTime domain: 7. SamplingFrequency domain: 7. SamplingNote: is a periodic function, consisting of a superposition of shifted replicas of X(j). Comparison of the DTFT of xk under different sampling period T 7. SamplingComparison of the DTFT of xk under different sampling period T 7. Sampling (Aliasing)Comparison of the DTFT of xk underdifferent sampling period T, contd 7. SamplingSampling Theorem 7. Samplings 2m （or fs 2fm ） Let x(t) be a band-limited signal. If the highest frequency of x x( (t t) is ) is mm, then, then x(t) can be uniquely determined by its samples under the following condition: Sampling Period T：or：Here, 2m (or 2fm ) is called Nyquist rate. ( Minimum distortionless sampling frequency ) BIO Harry Nyquist, American physical scientist, was born in Sweden, 1889, died in Texas 1976. He made significant contribution to Information Theory. In 1907, he migrated to America, and studied at Betact University. In 1917, he received the Ph.D degree at physics from Yale. In 19171934, he worked at AT&T company, later worked at Bell Lab.in 1927, Nyquist presented the famous Nyquist sampling theory: if a band-limited CT signal is sampled with sampling frequency up to the threshold, the time function can be recovered perfectly from the samples. The sampling frequency is no less than 2 times bandwidth of the CT signal.System for sampling and reconstruction: 7. SamplingRecovery 7. SamplingImplement of samplingsampling period T (s)sampling angular frequency ws=2p/T (rad/s)sampling frequency fs =1/T (Hz) 7. Sampling From the time-frequency relations and sampling theorem, we know the corresponding Nyquist rate are x(2t) 4fm(Hz)；x(t)*x(2t)2fm(Hz)；x(t)x(2t) 6fm(Hz)。 7. SamplingExample:Example: x(t) is a real-valued and band-limited signal with the highest frequency fm Hz . Please determine the Nyquist rate for x(2t), x(t)*x(2t) and x(t)x(2t) respectively.Solution: Solution: Practical Applications of Sampling TheoremIn many practical applications, the signals are not band-limitedAnti-aliasing filter 7. Sampling Comparisons between Aliasing error and Comparisons between Aliasing error and Truncation errorTruncation errorPractical Applications of Sampling Theorem 7. SamplingSampling Effects under Different Sampling FrequenciesSampling Frequency fs=44,100 HzSampling Frequency fs=5,512 HzSampling Frequency fs=5,512 Hz(using Anti-aliasing filtering before sampling) 7. SamplingConclusionsConclusions(1) Sampling in time domain will result in periodicity in frequency domain. The spectra of DT signal xk is the periodicity of CT signal x(t)s spectra.(2) Nyquist sampling theory: if a CT signal is band-limited, and it is sampled with sampling frequency up to the threshold, the time function can be recovered perfectly from the samples. the sampling frequency is no less than 2 times bandwidth of the CT signal.(3) Engineering applications of Sampling: if the practical signal is not a band-limited signal, x(t) can be recovered by passing the CT signal through an anti-aliasing filter with cutoff frequency at f = fm . 7. SamplingDiscussionDiscussion(1)Based on Nyquist Sampling Theorem, when we sample a CT signal, the sampling frequency is no less than 2 times bandwidth of the CT signal. But in practical engineering, the sampling frequency is no less than 35 times bandwidth of the CT signal. Why?(2)If the highest frequency of the CT signal is unknown, how do we choose the signal sampling period T? 7. Samplingxp(t) is a signal composed of many impulses, which are relatively difficult to transmit. It is more convenient to generate the sampled signal in a form referred to as a zero-order hold.7.1.2 Sampling with a Zero-order Hold(1) Sampling system construction: 7. Sampling 7. SamplingNote: Zero-order hold as impulse-train sampling followed by an LTI system with a rectangular impulse response.0110延时延时T 7. Sampling(2) Signal Recovery 7. SamplingLowpass filterIf 00100 7. SamplingInterpolation: The fitting of a continuous signal to a set of samples .(1) Ideal interpolation 7. Sampling7.2 The Reconstruction of a Signal from Its Samples Using Interpolationis the impulse response to theInterpolation functionideal lowpass filter. 7. SamplingIf , we get 7. Sampling(2) Zero-order hold interpolationInterpolation function01Magnitude spectrum 7. Sampling 7. SamplingT1000 7. Sampling(3) Linear interpolationInterpolation function 7. SamplingResult of applying a first-order hold (T/3)Result of applying a first-order hold (T/3) 7. SamplingExample 1: Original signal 00Recovered signal0 7. SamplingUndersampling Aliasing7.3 The Effect of Undersampling: AliasingIf , we get 7. Samplingwhere we have a change in the sign of the phase Example 2: Original signal: Recovered signal:s=60s=30 7. Samplings=1.50s=1.20 7. SamplingIf , we get Let , we can see thatNote: The sampling frequency should be greater than twice the highest frequency in the signal, rather than greater than or equal to twice the highest frequency. 7. SamplingExample 3: Original signal: Sampling a sinusoidal signal at exactly twice its frequency is not sufficientPractical Applications of Aliasing: (higher frequencies are reflected into lower frequencies)1. Sampling Oscilloscope:2. Strobe Effect:Rotating discStrobe 7. Sampling 1 2 3 4 7. SamplingDiscrete-Time System 7. Sampling7.4 Discrete-Time Processing of Continuous-Time Signals Continuous-time signal processing can be viewed as the cascade of three operations 1. C/D Conversion: In time domain：In frequency domain:Conversion of impulse train to discrete-time sequence 7. Sampling( denotes the frequency of DTFT) 7. SamplingThe conversion from the impulse train sequence of samples to the discrete-time sequence of samples can be thought of as a normalization in time. Scaling of the time axis by 1/T will introduce a scaling of the frequency axis by T.0100 7. Sampling2. D/C Conversion: 7. SamplingConversion of discrete-time sequence to impulse train 3. Overall system for filtering a CT signal using a DT filter 7. SamplingC/DD/CImpulse train to discrete-time sequenceDiscrete-time sequence to impulse train In general case： 7. Sampling0or 0 Example：Digital Differentiator Band-limited differentiator:000 7. Sampling00 (Highpass) 7. Sampling0010 7. Sampling7.5 Sampling of Discrete-Time Signals 1. is band-limited with 2. 7. Sampling0Reconstruction of xn If ， 7. Sampling(interpolation)In frequency domain： 7. Sampling7.5.2 Discrete-Time Decimation and InterpolationDecimation:(downsampling)i.e.,0100 7. SamplingDiscrete-TimeDecimation:(down-sampling)001000 7. SamplingContinuous-TimeInterpolation:(up-sampling) 7. Sampling 7. SamplingDown-samplingOriginal signalUp-samplingDown-samplingMaximum possible down-samplingApplications of Decimation and InterpolationDecimationInterpolationExample： 7. Sampling000000 7. SamplingT0000 7. SamplingHalf-Sample Delay：Dual with sampling in time domain.0 7. Sampling Sampling in Frequency DomainIn time domain:频域离散化频域离散化 时域周期化（时域周期化（ ）In frequency domain： 7. SamplingSampling theorem in frequency domain:1. is time bounded by 2.i.e. 7. Sampling 在频域，从频谱的样本重建连续频谱时的在频域，从频谱的样本重建连续频谱时的频域频域时限内插时限内插过程是过程是以矩形窗的频谱作为内插函数以矩形窗的频谱作为内插函数实实现的。现的。 7. SamplingReconstruction of x(t): rectangular window in time domainProblems: 7.3 7.6 7.7 7.13 7.14 7.15 7.167.19 7.23 7.26 7.27 7.29 7.35 7.42 7.45 7. SamplingInterpolation functionIn frequency domain：