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EE354 : Communications System I

EE354 : Communications System I. Lecture 24: Sampling Quantization Aliazam Abbasfar. Outline. Sampling Quantization. Sampling and Quantization. Migration to digital systems Discrete Quantized Benefits Digital Processing Digital recording Digital transmission

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EE354 : Communications System I

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  1. EE354 : Communications System I Lecture 24: Sampling Quantization Aliazam Abbasfar

  2. Outline • Sampling • Quantization

  3. Sampling and Quantization • Migration to digital systems • Discrete • Quantized • Benefits • Digital Processing • Digital recording • Digital transmission • Messages and signals are sampled and quantized • Minimum distortion • Pulse-coded modulation (PCM) systems

  4. Sampling • Ideal sampling y(t) = combTs(x) = Sx(nTs) d (t-nTs)  Y(f) = 1/T SX(f-n/Ts)= fsSX(f-nfs) • Distortion-less recovery if band-limited • fs >= 2W • W : maximum frequency component of the signal • Pre-sampling filter • Avoids fold-over distortion

  5. Signal reconstruction • If x(t) is band-limited to (1/2Ts) x(t) = Sx(nTs) sinc(t/Ts - n) • Discrete sequence x[n] represent a band-limited signal • x[n] = x(nTs) • Px = E[ x2[n] ]

  6. Quantization • Rounding the signal samples • Reducing signal levels • Quantization function • Decision levels • Quantized levels • # of quantized levels • Clipping level (Vc) • Dynamic range • Quantization error • eq = x - xq

  7. Quantization noise • Signals with quantization • x(t) = xq(t) + eq(t) • Quantization noise : eq(t) • Noise power • NQ= E[ eq2]= E[(x-xq)2] • Function of the PDF of x • Average of noise power in quantization bins • Design quantization law to minimize S/NQ • SQNR = S/NQ = E[x2]/E[(x-xq)2] • # of quantized levels • Uniform/non-uniform quantization

  8. Uniform PCM • Uniformly distributed x/ uniform quantizer • S = Vc2/3 • NQ = D2/12 • SQNR = 22n SQNR (dB) = 6.02 n dB • SQNR is less for signals with lower power

  9. Non-uniform PCM • m-law (US) • A-law (Europe)

  10. Uniform vs Non-uniform PCM • SQNR increases with signal power in uniform PCM • Almost flat in a wide range in non-uniform • Uniform PCM needs more bits to meet the specs

  11. ADC and DAC • Analog to digital Convertors (ADC) • Sampler + Quantizer + Coder • Produces digital values • Digital to analog Convertors (DAC) • Produces analog voltage • Reconstruct signals

  12. Reading • Carlson Ch. 12.1 • Proakis & Salehi ch. 6.6

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