ECE 4371, Fall, 2014 Introduction to Telecommunication Engineering/Telecommunication LaboratoryOutlineClaude Elwood Shannon, Harry NyquistSampling TheorySampling Block DiagramImpulse SamplingImpulse Sampling with increasing sampling time TIntroductionMathMath, cont.Interpolation FormulaInterpolationPractical InterpolationSampling TheoremUnder Sampling, AliasingAvoid AliasingAnti-AliasingAliasingExample: Aliasing of Sinusoidal SignalsSlide 20Slide 21Slide 22Slide 23Natural sampling (Sampling with rectangular waveform)Bandpass SamplingSlide 26Bandpass Sampling TheoremECE 4371, Fall, 2014Introduction to Telecommunication Engineering/Telecommunication Laboratory Zhu HanDepartment of Electrical and Computer EngineeringClass 7Sep. 17th, 2014OutlineOutlineAnalog vs. DigitalADC/DAC: gateway between analog and digital domains–Sampling Theorem–Quantization–Most important part in communication system–Most important during interview–Read books carefullyExamplesClaude Elwood Shannon, Harry NyquistClaude Elwood Shannon, Harry NyquistSampling TheorySampling TheoryIn many applications it is useful to represent a signal in terms of sample values taken at appropriately spaced intervals.The signal can be reconstructed from the sampled waveform by passing it through an ideal low pass filter.In order to ensure a faithful reconstruction, the original signal must be sampled at an appropriate rate as described in the sampling theorem. –A real-valued band-limited signal having no spectral components above a frequency of B Hz is determined uniquely by its values at uniform intervals spaced no greater than seconds apart.12B� �� �� �Sampling Block DiagramSampling Block DiagramConsider a band-limited signal f(t) having no spectral component above B Hz.Let each rectangular sampling pulse have unit amplitudes, seconds in width and occurring at interval of T seconds.A/D conversion f(t)Tfs(t)SamplingImpulse SamplingImpulse SamplingSignal waveform01 201Impulse sampler01 201Sampled waveform01 201Impulse SamplingImpulse Samplingwith increasing sampling time Twith increasing sampling time TSampled waveform01 201Sampled waveform01 201Sampled waveform01 201Sampled waveform01 201Introductionrate sampling:1 period sampling : where(3.1) )( )()( signal sampled ideal thedenote )(Let ssssnsTfTnTtnTgtgtgEquation number is not the same as in the bookMathMathnsmmssssnsmssmssmssnsW n fjWngfGWTWffGmffGffGffG nf TjnTgfGmffGftgmffGfTmfTfGnTtt(3.4) )exp()2()( 21 and for 0)( If(3.5) )()()(or (3.3) )2exp()()( obtain to(3.1) on Transformier apply Fourmay or we(3.2) )()( )()(1)()()g( have weA6.3 Table From0 Math, cont.Math, cont.)( of ninformatio all contains )2(or for )2(by determineduniquely is )((3.7) , )exp()2(21)(as )( rewritemay we(3.6) into (3.4) ngSubstituti(3.6) , )(21)( that (3.5) Equationfrom find we2.2 for 0)(.1 WithtgWngnWngtgWfWWnfjWngWfGfGWfWfGWfGWfWffGnsInterpolation FormulaInterpolation Formula)( offormula ioninterpolat an is (3.9)(3.9) - , )2(sin)2( 2)2sin()2( (3.8) )2(2exp 21)2( )2exp()exp()2(21 )2exp()()( havemay we, )2( from )(t reconstruc TotgtnWtcWngn Wtn WtWngdfWnt fjWWngdf f tjW n fjWngWdfftjfGtgWngtgnnnWWWWnInterpolationInterpolationnsssTnTtcnTgtg sin)()(nsssTnTtcnTgtg sin)()(If the sampling is at exactly the Nyquist rate, then)(tgPractical InterpolationPractical InterpolationSinc-function interpolation is theoretically perfect but it can never be done in practice because it requires samples from the signal for all time. Therefore real interpolation must make some compromises. Probably the simplest realizable interpolation technique is what a DAC does.)(tgSampling TheoremSampling Theoremrate. samplinghigher haveor bandwidth signal limit themay wealiasing, avoid .To occurs aliasingsampling)(under limited-bandnot is signal theWhen21 intervalNyquist 2 rateNyquist )2( from recovered completely be can signal The.2.)2(by described completely be can , tolimited is whichsignal1.a signals limited-bandstrictly for Theorem SamplingWWWngWngWfWUnder Sampling, AliasingUnder Sampling, AliasingAvoid AliasingAvoid AliasingBand-limiting signals (by filtering) before sampling.Sampling at a rate that is greater than the Nyquist rate.A/D conversionf(t)Tfs(t)SamplingAnti-aliasingfilterAnti-AliasingAnti-AliasingAliasingAliasing2D exampleExample: Aliasing of Sinusoidal SignalsExample: Aliasing of Sinusoidal SignalsFrequency of signals = 500 Hz, Sampling frequency = 2000HzExample: Aliasing of Sinusoidal SignalsExample: Aliasing of Sinusoidal SignalsFrequency of signals = 1100 Hz, Sampling frequency = 2000HzExample: Aliasing of Sinusoidal SignalsExample: Aliasing of Sinusoidal SignalsFrequency of signals = 1500 Hz, Sampling frequency = 2000HzExample: Aliasing of Sinusoidal SignalsExample: Aliasing of Sinusoidal SignalsFrequency of signals = 1800 Hz, Sampling frequency = 2000HzExample: Aliasing of Sinusoidal SignalsExample: Aliasing of Sinusoidal SignalsFrequency of signals = 2200 Hz, Sampling frequency = 2000HzNatural samplingNatural sampling(Sampling with rectangular waveform)(Sampling with rectangular waveform)Signal waveform01 201 401 601 801 1001 1201 1401 1601 1801 2001Natural sampler01 201 401 601 801 1001 1201 1401 1601 1801 2001Sampled waveform01 201 401 601 801 1001 1201 1401 1601 1801 2001Figure 6.10Bandpass SamplingBandpass Sampling(a) variable sample rate(b) maximum sample rate without aliasing(c) minimum sampling rate without aliasingBandpass SamplingBandpass SamplingA signal of bandwidth B, occupying the frequency range between fL and fL + B, can be uniquely reconstructed from the samples if sampled at a rate fS : fS >= 2 * (f2-f1)(1+M/N) where M=f2/(f2-f1))-N and N = floor(f2/(f2-f1)), B= f2-f1, f2=NB+MB.Bandpass Sampling TheoremBandpass