1Video TransmissionVID3: Sampling and QuantizationBy Prof. Gregory D. Durgincopyright 2009 – all rights reservedClaude E. Shannon (1916-2001) Mathematician and Electrical Engineer Worked for Bell LabsffAlumnus of U of Michigan and MIT Eccentric researcher “Father of Information Theory” Key Paper“A Mathematical Theory of Communication”2A Mathematical Theory of Communication Published in Bell System Technical Journal, 1948.2Key Contributions of Shannon’s Paper Rate Distortion Theorem Given a distortion criterion, what is the minimum number of bits for representing a signal?number of bits for representing a signal? Lossless Data Compression What is the minimum number of irreducible bits that can reproduce a data set? Channel Capacity Theorem3 What is the minimum rate (in bits/sec) that data can be sent across a noisy channel? Bonus: Nyquist Sampling TheoremQuick Review of Fourier Transforms…Impulse TrainImpulse TrainSinc FunctionBox Function43Sampling TheoremTime DomainOriginal Signal......with Band-limited SpectrumFrequency Domain Start with time-domain signal with band-limited spectrumtime, tfreq, f5 Continuous, real-valued signalImpulsive SamplingTime DomainUniform ReplicationImpulsive SamplingFrequency Domain Impulsive Sampling: keep only finite number of samples Multiply by impulse train in the time domainfreq, ftime, t6 Convolve with impulse train in the frequency domain Mixing and/or filtering will recover exact original signal Requires sampling is faster than 2fmax(Nyquist rate)4Non-Impulsive SamplingTime DomainNon-Uniform ReplicationNon-Impulsive SamplingFrequency Domain Time-Domain: multiply by a square waveFreq-Domain: convolve with non-uniform impulse trainfreq, ftime, t7FreqDomain: convolve with nonuniform impulse train Nyquist rate still applies or else aliasing resultsSample-and-HoldTime DomainSinc-Conforming ReplicationSample-and-HoldFrequency Domain Time Domain sequence: impulsive sample, convolve with square pulsefreq, ftime, t8 Frequency Domain sequence: convolve signal spectrum with impulse train, multiply by sinc envelope Precise signal recovery requires filtering & equalization5Pulse Amplitude Modulation (PAM)Time DomainNon-Impulsive SamplingPAM with Triangular PulsesFrequency Domain Time Domain: impulsive sample, convolve with arbitrary pulse shapefreq, ftime, t9 Frequency Domain: convolve signal spectrum with impulse train, multiply by single-pulse spectral envelope Precise signal recovery requires filtering & equalizationExample: Audio Spectral ContentFull Spectrum20kHz Spectrum33.5x 107Filter Levels20kHz Spectrum10kHz Spectrum5kHz Spectrum0511.522.5Aliasing10-3 -2 -1 0 1 2 3x 10400.5Hz10% aliasing100% aliasing6Quantization11Example of Uniform Signal QuantizationQuantization Noise127Quantization Noise8-bit quantization (SNR of 48 dB)Quantization Levels6-bit quantization (SNR of 36 dB)4-bit quantization (SNR of 24 dB)3-bit quantization (SNR of 18 dB)2-bit quantization (SNR of 12 dB)13q( )1-bit quantization (SNR of 6 dB)Properties of Uniform Quantization Provides uniformly-spaced quantization levels Can span [0,+Vmax] or [-Vmax,+Vmax]f Works best when signal levels uniformly distributed Typical # of levels is related to # of bits/sample 2^M, where M is number of bits/sample Spacing of [Vmax –Vmin]/(2^M - 1)14pg[]() Quantization Signal-to-Noise Ratio is 6M (ideal) Non-uniform signals must be companded8Companding a SignalVVinVoutt15tNon-Uniform Quantization Alternative to companding Requires Lloyd-Max algorithmf Iterative procedure for choosing optimal levels Could use results to design optimal compander as well169Vector Quantization Quantize 2 or more samples simultaneously Only way to approach Shannon Rate Distortion limitVsample 2Vsample 2Vsample 1Vsample 117Conventional SchemeVector QuantizationExample: Digitizing Analog Video Baseband signal has 5 MHz maximum frequency Remember: starting point is a lousy analog signal/ Nyquist sampling rate is 10 Msamples/sec Let’s assume 8-level quantization Requires 8 bits/sample Visible SNR of 48 dB – pretty good pictureRequires uncompressed bit rate of 80 Mbits/sec18Requires uncompressed bit rate of 80 Mbits/sec Way too fast for many wired connections Signal is still poor analog video plus quantization