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蔡宗珉 : Multi-stage Filter Implementation
Class Report 蔡宗珉 : Multi-stage Filter Implementation
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Filter Design Pass Band Transition Band 1+ p 1- p Stop Band s p s p p = ripple (deviation) in the passband from the ideal response s = ripple (deriation) in the stopband from the ideal response p = passband edge frequency s = stopband edge frequency = p = s = p - s F = fp - fs
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High Decimation Conversion
LPF 100 10,000 Hz 100 Hz N 5 45 50 5000 F = 50 – 45 = 5 Hz F = 10,000 Hz = 2F/F = = 0.001 N 7250 Total multiplications per second:
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Multi-stage Decimation
LPF1 LPF2 50 2 10,000 Hz 200 Hz 100 Hz N2 N1 45 150 5000 45 50 100 F = 150 – 45 = 105 F = 50 – 45 = 5 N2 150 N1 345 R1 34,500 R2 7,500 Total multiplications per second: R = R1 + R2 = 42,000
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Design of practical sampling rate converters
Specify the filter requirements and those for individual stages Determine the optimum number of stages of decimation or interpolation Determine the decimation or interpolation factors for each stage Design an appropriate filter for each stage
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Multi-stage Interpolation
y[m] x[n] h[m] L F FI x[n] y[m] L1 h1(m1) L2 h2[m2] LI hI[mI]
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Multi-stage Decimation
y[m] x[n] Number of multiplications: M h[m] F0 FI x[n] y[m] h1(m1) M1 h2(m2) M2 hI(mI) MI FI-1 F0 FI F1 Number of multiplications:
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Filter Requirements for Individual Stages
y[m] x[n] h1(m1) M1 h2(m2) M2 hI(mI) MI Overall filter requirement for decimation Passband 0 f fp Stage k stopband Fk - fs f Fk-1/2 k = 1, 2, … , I - 1 Last stage stopband fs f FI-1/2 Passband ripple p/I Stopband ripple s
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GSM Digital Down Converter
CIC filter Half band filter Linear phase filter I x(n) cos C(z) CIC Decimation 48:1 D(z) Decimation 2:1 E(z) Decimation 2:1 DDS Q sin KHz 52 MHz 1.08 MHz KHz Frequency Characteristics: H(ej) = C(ej) D(ej) E(ej) C(ej) D(ej) E(ej) H(ej) -50 -20 -50 -50 -100 -40 -100 -100 -150 -60 0.5 0.5 0.5 500 KHz
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CIC Filter CIC Filter: Cascaded Integrator Comb Filter
Purpose: Large sample rate changes of digital systems. Advantages: No multiplier required No memory for coefficients required Low power consumption Very regular structure with two simple basic block Little external control required
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CIC Building Block Basic Integrator: A single pole IIR filter with a unity feedback y[n] = y[n - 1] + x[n] Basic Comb: y[n] = x[n] – x[n – M] Filter Structure: Z-1 Z-M + - + + Integrator Element Comb Element
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Half Band Filter The attractive properties is about 50% of the
coefficients are zero. The half band filter can be designed with linear phase under a constraint on its length. Thus, the required number of multiplications can be reduced to about 25%. The pass band and stop band ripples are equal, i.e. p = s. The past band and stop band edges are symmetric with respect to /2, i.e. p + s = .
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