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@@ -69,7 +69,7 @@ where \\(F\\) is the ratio of single-stage low pass filter's transition region w
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F = \frac{f\_{\text{stop}} - B^\prime}{f\_{\text{stop}}}
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\end{equation}
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After using Eq. \ref{eq:M1opt} to determine the optimum first downsampling factor, and setting \\(M\_1\\) equal to the integer sub-multiple of \\(M\\) that is closest to \\(M\_{1,\text{opt}}\\), the second downsampling factor is:
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After using Eq. \eqref{eq:M1opt} to determine the optimum first downsampling factor, and setting \\(M\_1\\) equal to the integer sub-multiple of \\(M\\) that is closest to \\(M\_{1,\text{opt}}\\), the second downsampling factor is:
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\begin{equation} \label{eq:M2\_from\_M1}
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M\_2 = \frac{M}{M\_1}
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@@ -89,9 +89,9 @@ N = \frac{\text{Atten}}{22 (f\_{\text{stop}} - f\_{\text{pass}})} = \frac{60}{22
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which is way too large for practical implementation.
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To reduce the number of necessary filter taps, we can partition the decimation problem into two stages.
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With \\(M = 100\\), \\(F = (2200-1800)/2200\\), Eq. \ref{eq:M1opt} yields \\(M\_{1,\text{opt}} = 26.4\\).
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With \\(M = 100\\), \\(F = (2200-1800)/2200\\), Eq. \eqref{eq:M1opt} yields \\(M\_{1,\text{opt}} = 26.4\\).
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The integer sub-multiple of 100 closest to \\(26.4\\) is \\(25\\), so we set \\(M\_1 = 25\\).
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Next, from Eq. \ref{eq:M2\_from\_M1}, \\(M\_2 = 4\\) is found.
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Next, from Eq. \eqref{eq:M2\_from\_M1}, \\(M\_2 = 4\\) is found.
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The first low pass filter has a pass-band cutoff frequency of \\(1.8\\,kHz\\) and its stop-band is \\(400/25 - 1.8 = 14.2\\,kHz\\) ([Figure 4](#figure--fig:decimation-two-stage-example) (d)).
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The second low pass filter has a pass-band cutoff frequency of \\(1.8\\,kHz\\) and its stop-band is \\(4-1.8 = 2.2\\,kHz\\).
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@@ -116,16 +116,18 @@ There are two **practical issues** to consider for two-stage decimation:
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In order to cope with the second issue, an efficient decimation filter implementation scheme called _polyphase decomposition_ can be used.
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<summary>The advantages of two stage decimation, over single-stage decimation are:
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<div class="sum">
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<ul class="org-ul">
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<li>an overall reduction in computation workload</li>
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<li>reduced signal and filter coefficient data storage</li>
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<li>simpler filter designs</li>
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<li>a decrease in the ill effects of finite binary-work-length filter coefficients</li>
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</ul>
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The advantages of two stage decimation, over single-stage decimation are:
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These advantages become more pronounced as the overall desired decimation factor \(M\) becomes larger.</summary>
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- an overall reduction in computation workload
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- reduced signal and filter coefficient data storage
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- simpler filter designs
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- a decrease in the ill effects of finite binary-work-length filter coefficients
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These advantages become more pronounced as the overall desired decimation factor \\(M\\) becomes larger.
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</div>
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### References: {#references}
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