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. Fix-rate Signal Processing Fix rate filters - same number of input and output samples Filter x(n) 8 samples y(n) 8 samples y(n) = h(n) * x(n) Figure 1
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. Multirate Signal Processing Multirate filters - different numbers of input and output samples Filter x(n) 8 samples y(n) 4 samples Figure 2
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. Multirate Signal Processing Filter Decimation - M>NM samplesN samples Filter Interpolation - M<NM samplesN samples Figure 3
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Decimator x(n)x(n) M y(n)y(n) Basic application - reduce bit-rate by discarding samples Consequence - Distortion and Aliasing error Figure 4
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Interpolator x(n)x(n) M y(n)y(n) Basic application - Insert samples between missing gaps Consequence - Restore the number of samples before decimation Figure 4
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Decimation x(n)x(n) M y(n)y(n) -4-3-201234-4-3-201234 e.g. M = 2x(n)x(n)x’(n) -2012 y(n)y(n) Figure 5
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Decimation x(n)x(n)x’(n) x’(n) = x(n) n = 0, +M, +2M, +3M, +4M,... 0 otherwise (1)
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Decimation (1) i(n) is a periodic impulse train that can be expressed as (2) i(n)i(n) 0M2M-2M-M Figure 6
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Decimation (1) (2) (3)
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Decimation x(n)x(n) M y(n)y(n) -4-3-201234-4-3-201234 e.g. M = 2x(n)x(n)x’(n) -2012 y(n)y(n)
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Decimation x(n)x(n)x’(n)y(n)y(n) y(n) = x’(Mn)(4) According to equation (3) hence (5)
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Decimation Distortion due to Decimation can be seen in the frequency domain Consider the z transform of x(n) and x’(n) (6) According to equation (3)
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Decimation According to equation (3) (7)
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Decimation According to equation (4),y(n) = x’(Mn) (8)where p = Mn (9)
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Key equations of Decimation x(n)x(n)x’(n)y(n)y(n) y(n) = x’(Mn)
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Key equations of Decimation x(n)x(n)x’(n)y(n)y(n) Convert to DFT with z = e j z transform
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Spectral Changes in Decimation x(n)x(n)x’(n)y(n)y(n) 0 /4 /4 /2 /4 /4 /2 /4 /4 /2 /4 Figure 7a
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Spectral Changes in Decimation x(n)x(n)x’(n)y(n)y(n) 0 /4 /4 /2 /4 /4 /2 /4 /4 /2 /4 0 /4 /4 /2 /4 /4 /2 /4 /4 /2 /4 images M=4 Figure 7a Figure 7b
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Spectral Changes in Decimation x(n)x(n)x’(n)y(n)y(n) 0 /4 /4 /2 /4 /4 /2 /4 /4 /2 /4 0 /4 /4 /2 /4 /4 /2 /4 /4 /2 /4 0 /4 /4 /2 /4 /4 /2 /4 /4 /2 /4 M=4 Figure 7a Figure 7b Figure 7c
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Spectral Changes in Decimation x(n)x(n)x’(n)y(n)y(n) 0 /4 /4 /2 /4 /4 /2 /4 /4 /2 /4 0 /4 /4 /2 /4 /4 /2 /4 /4 /2 /4 0 /4 /4 /2 /4 /4 /2 /4 /4 /2 /4 M=4 Figure 7a Figure 7b Figure 7c
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Spectral Changes in Decimation x(n)x(n)x’(n) 1. Retaining one out of M samples in x(n) generates M replicated images of the original spectrum. 2. The spacing of images is 2 /M 3. Decimation by a factor of M stretches the width of the spectrum by M times x’(n)y(n)y(n)
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Interpolation x(n)x(n) M y(n)y(n) -4-3-201234 e.g. M = 2x’(n) -2012 y(n)y(n) -4-3-201234 e.g. M = 3x’(n) -2012 y(n)y(n) -556-6 Figure 8
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Interpolation x(n)x(n) M y(n)y(n) (10) y(n) = x(n/M) n = 0, +M, +2M, +3M,... 0 otherwise (11) (12)
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Spectral Changes in Interpolation x(n)x(n) M y(n)y(n) (13)
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Spectral Changes in Interpolation 0 /4 /4 /2 /4 /4 /2 /4 /4 /2 /4 Figure 9a x(n)x(n) M y(n)y(n) 0 /4 /4 /2 /4 /4 /2 /4 /4 /2 /4 M=4 Figure 9b
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Spectral Changes in Interpolation 0 /4 /4 /2 /4 /4 /2 /4 /4 /2 /4 Figure 9a x(n)x(n) M y(n)y(n) 0 /4 /4 /2 /4 /4 /2 /4 /4 /2 /4 M=4 Figure 9b
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Spectral Changes in Interpolation x(n)x(n)y(n)y(n) 2. M images separating from each other by a spacing of 2 /M are generated 1. Interpolation by a factor of M compresses the width of the spectrum by M times
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Decimation & Interpolation (M=4) -4-3-201234 -2012 -4-3-201234 M M Input Sequence Output Sequence Figure 10 56-6-5 -4-3-20123456-6-5
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Decimation & Interpolation (M=4) 0 /4 /4 /2 /4 /4 /2 /4 /4 /2 /4 M=4 M M Bandwidth - /8 Figure 11
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Decimation & Interpolation It seems that the bitrate can be reduced simply by decimation
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Decimation & Interpolation But something is wrong,what’s the problem?
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