Difference between revisions of "Text slecture" - Rhea

Revision as of 20:14, 9 October 2014

Downsampling

A slecture by ECE student Yerkebulan Yeshmukhanbetov

Partly based on the ECE438 Fall 2014 lecture material of Prof. Mireille Boutin.

Outline

Introduction
Definition of Downsampling
Derivation of DTFT of downsampled signal
Example
Conclusion
References

@@ Line 89: / Line 89: @@
 == Introduction  ==
-This slecture provides definition of downsampling, derives DTFT&nbsp; downsampled signal and demonstrates it in a frequency domain. Also, it explains process of decimation and why it needs a low-pass filter.
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-== Definition of Downsampling<br>  ==
-Downsampling is an operation which involves throwing away samples from discrete-time signal. Let&nbsp; ''x[n]'' be a digital-time signal shown below: <br>
-GRAPH<br>
-&nbsp;then y[n] will be produced by downsampling ''x [n]''&nbsp; by factor ''D'' = 3. So, ''y [n] = x[Dn]''.
-GRAPH<br>
-As seen in above graph, ''y [n]'' is obtained by throwing away some samples from x [n]. So, ''y [n]'' is a downsampled signal from
-''x [n]''.<br>
 ----
-== Derivation of DTFT&nbsp;of downsampled signal  ==
+== Definition   ==
-Let ''x (t) ''be a continuous tim''e ''signal. Then ''x<sub>1</sub> [n] = x (T<sub>1</sub>n) ''and''&nbsp; x<sub>2</sub> [n] = x (T<sub>2</sub>n)''. And ratio of sampling periods would be
-D = T<sub>2</sub>/T<sub>1</sub>, &nbsp; which is an integer greater than 1. From these equations we obtain realtionship between ''x<sub>1</sub> [n]'' and ''x<sub>2</sub> [n]''. <br>
-<math>\begin{align}
-x_2 [n] = x(T_2 n) = x(DT_1 n) = x_1 [nD]
-\end{align}</math>
-Below we derive Discrete-Time Fourier Transform of ''x<sub>2</sub> [n]'' in terms of DTFT of ''x<sub>1</sub> [n]''.
-<br>
+== Derivation l  ==
-<math>\begin{align}
-&\mathcal{X}_2(\omega)= \mathcal{F}(x_2 [n]) = \mathcal{F}(x_1 [Dn])\\
-&=  \sum_{n = -\infty}^\infty x_1[Dn] e^{-j \omega n} =  \sum_{m = -\infty}^\infty x_1[m] e^{-j \omega {\frac{m}{D}}}\\
-&=  \sum_{n = -\infty}^\infty s_D[m]* x_1 [m] e^{-j \omega {\frac{m}{D}}}\\
-\end{align}</math>
-<br>
-where <br>
-<math>s_D [m]=\left\{ \begin{array}{ll}
-,& \text{ if } n \text{ is a multiple of } 4,\\
-, & \text{ else}.
-\end{array}\right. = {\frac{1}{D}} \sum_{k = -\infty}^{D-1} e^{jk {\frac{2 \pi}{D} m}}</math>
-<br>
-<math>\begin{align}
-&\mathcal{X}_2(\omega)= \sum_{m = -\infty}^\infty {\frac{1}{D}} \sum_{k = -\infty}^{D-1} e^{jk {\frac{2 \pi}{D} m}}  x_1[m] e^{-j \omega {\frac{m}{D}}}\\
-&= {\frac{1}{D}} \sum_{k = -\infty}^{D-1} \sum_{m = -\infty}^\infty  x_1[m] e^{-jm ({\frac{\omega - 2 \pi k}{D}})} =  \\
-&=  {\frac{1}{D}} \sum_{k = -\infty}^{D-1} \mathcal{X}_1 ({\frac{\omega - 2 \pi k}{D}}) \\
-\end{align}</math>
-----
 == Example<br>  ==
@@ Line 152: / Line 106: @@
 <br>
-Let's take a look&nbsp; at&nbsp; an original signal ''X<sub>1</sub> (w)'' and&nbsp;&nbsp;''X<sub>2</sub> (w)'' which is obtained after downsampling X<sub>1</sub>(w) by factor D = 2 in a frequency domain.
-[[Image:Downsamplegraph.jpg]]<br>
-<br>
-<br>
-<br>
-We would like to determine what <span class="texhtml">''X''<sub>''s''</sub>(''f'')</span> looks like in order to find a way to reconstruct <span class="texhtml">''x''(''t'')</span>.
-Since we have sampled at a rate <span class="texhtml">''f''<sub>''s''</sub> &gt; 2''f''<sub>''M''</sub></span>, the following inequalities hold:
-<math> f_s > 2f_M \iff f_s - f_M > f_M \iff -f_s + f_M < f_M </math>.
-Together, these inequalities, the graph of <span class="texhtml">''X''(''f'')</span>, and the expression for <span class="texhtml">''X''<sub>''s''</sub>(''f'')</span> in terms of <span class="texhtml">''X''(''f'')</span> imply that <span class="texhtml">''X''<sub>''s''</sub>(''f'')</span> will have the spectrum shown in the figure below.
-[[Image:Xs1.png]]
-Notice that the spectrum of the ideal sampling of a signal is an amplitude scaled periodic repetition of the original spectrum. Since <span class="texhtml">''x''(''t'')</span> is bandlimited and we have sampled at a rate <span class="texhtml">''f''<sub>''s''</sub> &gt; 2''f''<sub>''M''</sub></span>, the periodic repetitions of <span class="texhtml">''X''(''f'')</span> do not overlap.
-All the information needed to reconstruct <span class="texhtml">''X''(''f'')</span> can be found in the portion of <span class="texhtml">''X''<sub>''s''</sub>(''f'')</span> that corresponds to <span class="texhtml">''X''(''f'')</span> (shown in red). Therefore we can use a simple lowpass filter with gain <math> \tfrac{1}{f_s} </math> and cutoff frequency <math> \tfrac{f_s}{2} </math> to recover <span class="texhtml">''X''(''f'')</span> from <span class="texhtml">''X''<sub>''s''</sub>(''f'')</span>.
-<br> <math>
-X(f) = X_s(f)\left\{
-\begin{array}{ll}
-\frac{1}{f_s}, & |f| \le \frac{f_s}{2}\\
-, & \text{else}
-\end{array}
-\right. </math>
-<math> \iff x(t) = x_s(t)*\text{sinc}(f_st) </math>
-<math> \therefore </math> We can perfectly reconstruct <span class="texhtml">''x''(''t'')</span> from <span class="texhtml">''x''<sub>''s''</sub>(''t'')</span>.
 ----
@@ Line 192: / Line 112: @@
 == Example  ==
-Though the Nyquist theorem states that perfect reconstruction is possible if we satisfy the Nyquist condition <span class="texhtml">(''f''<sub>''s''</sub> &gt; 2''f''<sub>''M''</sub>)</span>, it is important to note that this condition is not necessary. The following example demonstrates how perfect reconstruction is sometimes possible even when undersampling.
-Let the signal <span class="texhtml">''x''(''t'')</span> have a spectrum <span class="texhtml">''X''(''f'')</span> as seen in the figure below.
-[[Image:X2.png]]
-The Nyquist condition states that we should sample at a rate <span class="texhtml">''f''<sub>''s''</sub> &gt; 2(2''a'') = 4''a''</span>. Instead, let us sample at <span class="texhtml">''f''<sub>''s''</sub> = 2''a''</span>.
-As before, we have <math> x_s(t) = x(t)p_{\frac{1}{f_s}}(t) </math> and <math> X_s(f) = f_s\sum_{k = -\infty}^\infty X(f-kf_s) </math>
-<math> \implies X_s(f) = 2a\sum_{k = -\infty}^\infty X(f-2ka) </math>.
-Therefore, <span class="texhtml">''X''<sub>''s''</sub>(''f'')</span> will have the spectrum shown in the figure below.
-[[Image:Xs2.png]]
-Notice that there is no aliasing in <span class="texhtml">''X''<sub>''s''</sub>(''f'')</span> even though <span class="texhtml">''f''<sub>''s''</sub> &lt; 4''a''</span>. In addition, the portion of <span class="texhtml">''X''<sub>''s''</sub>(''f'')</span> that corresponds to <span class="texhtml">''X''(''f'')</span> (shown in red) can be recovered using a bandpass filter with gain <math> \tfrac{1}{2a} </math> and cutoff frequencies <span class="texhtml">''a'' and 2''a''</span>.
 ----
@@ Line 276: / Line 179: @@
 <font size="3"></font>
-<font size="3">To summarize, the Nyquist theorem states that any bandlimited signal can be perfectly reconstructed from its sampling if sampled at a rate greater than twice its bandwidth <span class="texhtml">(''f''<sub>''s''</sub> &gt; 2''f''<sub>''M''</sub>)</span>. However, the Nyquist condition is not necessary for perfect reconstruction as shown in the example above. </font>
 <font size="3"></font>
@@ Line 284: / Line 185: @@
 == References  ==
-[1] John G. Proakis, Dimitris G. Manolakis, "Digital Signal Processing with Principles, Algorithms, and Applications" 4th Edition,2006
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-== [[Nyquist Miguel Castellanos ECE438 slecture review|Questions and comments]]  ==
-If you have any questions, comments, etc. please post them on [[Nyquist Miguel Castellanos ECE438 slecture review|this page]].
-----
-[[2014 Fall ECE 438 Boutin|Back to ECE438, Fall 2014]]
-[[Category:Slecture]] [[Category:ECE438Fall2014Boutin]] [[Category:ECE]] [[Category:ECE438]] [[Category:Signal_processing]]

Difference between revisions of "Text slecture" - Rhea

Revision as of 20:14, 9 October 2014

Contents

Outline

Introduction

Definition

Derivation l

Example

Example

Conclusion

References

Alumni Liaison