MIT16_36s09_lec04

# MIT16_36s09_lec04 - MIT OpenCourseWare http://ocw.mit.edu...

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Unformatted text preview: MIT OpenCourseWare http://ocw.mit.edu 16.36 Communication Systems Engineering Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms . Lecture 4: Quantization Eytan Modiano A ero- A stro Dept. Eytan Modiano Slide 1 Sampling • Sampling provides a discrete-time representation of a continuous waveform – Sample points are real-valued numbers – In order to transmit over a digital system we must first convert into discrete valued numbers Quantization levels Q 3 Q 2 Q 1 λ λ λ λ λ λ λ λ λ λ Sample points What are the quantization regions What are the quantization levels Eytan Modiano Slide 2 Uniform Quantizer Δ 2 Δ 3 Δ − 3 Δ − 2 Δ −Δ 000 001 010 011 100 101 110 111 Y = x Q(x) 000 001 010 000 001 011 010 000 001 100 011 010 000 001 101 100 011 010 000 001 110 101 100 011 010 000 001 111 110 101 100 011 010 000 001 Q(x) • All quantization regions are of equal size ( Δ ) – Except first and last regions if samples are not finite valued • With N quantization regions, use log 2 (N) bits to represent each quantized value Eytan Modiano Slide 3 Quantization Error e(x) = Q(x) - x Squared error: D = E[e(x) 2 ] = E[(Q(x)-x) 2 ] SQNR: E[X 2 ]/E[(Q(x)-x) 2 ] Eytan Modiano Slide 4 Example • X is uniformly distributed between -A and A – f(x) = 1/2A, -A<=x<=A and 0 otherwise • Uniform quantizer with N levels => Δ = 2A/N – Q(x) = quantization level = midpoint of quantization region in which x lies • D = E[e(x) 2 ] is the same for quantization regions D = E [ e ( x ) 2 | x ! R i ] = x 2 f ( x ) dx " # /2 # /2 \$ = 1 # x 2 dx "# /2 # / 2 \$ = # 2 12 E [ X ] = 1 2 A x 2 dx " A A \$ = A 2 3 SQNR = A 2 /3 # 2 /12 = A 2 /3 (2 A / N ) 2 /12 = N 2 , ( # = 2 A / N ) Eytan Modiano Slide 5 Quantizer design...
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## This note was uploaded on 11/07/2011 for the course AERO 16.38 taught by Professor Alexandremegretski during the Spring '09 term at MIT.

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MIT16_36s09_lec04 - MIT OpenCourseWare http://ocw.mit.edu...

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