(One intermediate revision by the same user not shown)
Line 32: Line 32:
 
&= e^{-j\frac{2\pi}{N}k} X_N[k], \;\;\; k=0,...,N-1 \\
 
&= e^{-j\frac{2\pi}{N}k} X_N[k], \;\;\; k=0,...,N-1 \\
 
\end{align}\,\!</math>
 
\end{align}\,\!</math>
 
 
  
 
<math>
 
\text{(c)} \quad y[n]=\left\{\begin{array}{ll}x[n/2], &  n \text{ is even},\\ 0, & n \text{ is odd},\end{array} \right. n=0,...,2N-1
 
</math>
 
 
<math>\begin{align}
 
{\color{White}abcde} Y_{2N}[k] &= \sum_{n=0, n\text{ even}}^{2N-1}x\left[\frac{n}{2}\right]e^{-j\frac{2\pi}{N}kn}, \;\;\; \text{Let } n=2m \\
 
&= \sum_{m=0}^{N-1}x[m]e^{-j\frac{2\pi}{N}k2m} \\
 
&= X_N[2k], \;\;\; k=0,...,2N-1 \\
 
\end{align}\,\!</math>
 
  
 
<math>
 
<math>
Line 55: Line 43:
 
&= \sum_{m=0}^{N-1}x[m]e^{-j\frac{2\pi}{2N}k2m} \\
 
&= \sum_{m=0}^{N-1}x[m]e^{-j\frac{2\pi}{2N}k2m} \\
 
&= \sum_{m=0}^{N-1}x[m]e^{-j\frac{2\pi}{N}km}, \;\;\; k=0,...,2N-1 \\
 
&= \sum_{m=0}^{N-1}x[m]e^{-j\frac{2\pi}{N}km}, \;\;\; k=0,...,2N-1 \\
&= \left\{\begin{array}{ll}\sum_{m=0}^{N-1}x[m]e^{-j\frac{2\pi}{N}km} , &  k=0,1,...,N-1,\\ \sum_{m=0}^{N-1}x[m]e^{-j\frac{2\pi}{N}km}, & k=N,N+1,...,2N-1,\end{array} \right. , \text{ Let } k'=k-N  \\
+
&= X_N[k], \;\;\; k=0,...,2N-1 \\
&= \left\{\begin{array}{ll}\sum_{m=0}^{N-1}x[m]e^{-j\frac{2\pi}{N}km} , &  k=0,1,...,N-1,\\ \sum_{m=0}^{N-1}x[m]e^{-j\frac{2\pi}{N}(k'+N)m}, & k=N,N+1,...,2N-1,\end{array} \right.  \\
+
&= \left\{\begin{array}{ll}\sum_{m=0}^{N-1}x[m]e^{-j\frac{2\pi}{N}km} , &  k=0,1,...,N-1,\\ \sum_{m=0}^{N-1}x[m]e^{-j\frac{2\pi}{N}k'm}, & k=N,N+1,...,2N-1,\end{array} \right.  \\
+
&= \left\{\begin{array}{ll}X_N[k] , &  k=0,1,...,N-1,\\ X_N[k'], & k=N,N+1,...,2N-1,\end{array} \right.  \\
+
&= \left\{\begin{array}{ll}X_N[k] , &  k=0,1,...,N-1,\\ X_N[k-N], & k=N,N+1,...,2N-1,\end{array} \right.  \\
+
 
\end{align}\,\!</math>
 
\end{align}\,\!</math>
  
Line 75: Line 59:
 
&= \frac{1}{2}\sum_{n=0}^{N-1} x[n]e^{-j\frac{2\pi}{N}kn} + \frac{1}{2}\sum_{n=0}^{N-1} e^{-j\frac{2\pi}{N}\left(\frac{N}{2}\right)n} x[n]e^{-j\frac{2\pi}{N}kn}, \;\;\; (e^{j\pi n}=(-1)^n) \\
 
&= \frac{1}{2}\sum_{n=0}^{N-1} x[n]e^{-j\frac{2\pi}{N}kn} + \frac{1}{2}\sum_{n=0}^{N-1} e^{-j\frac{2\pi}{N}\left(\frac{N}{2}\right)n} x[n]e^{-j\frac{2\pi}{N}kn}, \;\;\; (e^{j\pi n}=(-1)^n) \\
 
&= \frac{1}{2}X_N[k] + \frac{1}{2}X_N\left[k-\frac{N}{2}\right], \;\;\; k=0,...,\frac{N}{2}-1 \\
 
&= \frac{1}{2}X_N[k] + \frac{1}{2}X_N\left[k-\frac{N}{2}\right], \;\;\; k=0,...,\frac{N}{2}-1 \\
 +
 
\end{align}\,\!</math>
 
\end{align}\,\!</math>
  

Latest revision as of 19:22, 26 February 2012



Solution to Q3 of Week 7 Quiz Pool


$ \begin{align} \text{(a)} \quad & y[n]=e^{j\frac{2\pi}{N}n}x[n], \;\;\; n=0,...,N-1 \\ & X_N[k]=\sum_{n=0}^{N-1}x[n]e^{-j\frac{2\pi}{N}kn} \\ \end{align}\,\! $

$ \begin{align} {\color{White}abcde} Y_N[k] &= \sum_{n=0}^{N-1}e^{j\frac{2\pi}{N}n}x[n]e^{-j\frac{2\pi}{N}kn}, \;\;\; k=0,...,N-1 \\ &= \sum_{n=0}^{N-1}x[n]e^{-j\frac{2\pi}{N}(k-1)n} \\ &= X_N[k-1] \\ \end{align} $



$ \begin{align} \text{(b)} \quad & y[n]=\left\{\begin{array}{ll}x[N-1], & n=0,\\ x[n-1], & n=1,...,N-1\end{array} \right.\\ \end{align}\,\! $

$ \begin{align} {\color{White}abcde} Y_N[k] &= x[N-1] + \sum_{n=1}^{N-1}x[n-1]e^{-j\frac{2\pi}{N}kn}, \;\;\; \text{Let } m=n-1 \\ &= x[N-1] + \sum_{m=0}^{N-2}x[m]e^{-j\frac{2\pi}{N}k(m+1)} \\ &= x[N-1] + e^{-j\frac{2\pi}{N}k}\sum_{m=0}^{N-2}x[m]e^{-j\frac{2\pi}{N}km} \\ &= x[N-1] + e^{-j\frac{2\pi}{N}k}\sum_{m=0}^{N-1}x[m]e^{-j\frac{2\pi}{N}km} - e^{-j\frac{2\pi}{N}k}x[N-1]e^{-j\frac{2\pi}{N}k(N-1)} \\ &= x[N-1](1-e^{-j2\pi k}) + e^{-j\frac{2\pi}{N}k}\sum_{m=0}^{N-1}x[m]e^{-j\frac{2\pi}{N}km}, \;\;\; ( e^{-j2\pi k} = 1, \; \forall \; \text{integer} \; k ) \\ &= e^{-j\frac{2\pi}{N}k}\sum_{m=0}^{N-1}x[m]e^{-j\frac{2\pi}{N}km} \\ &= e^{-j\frac{2\pi}{N}k} X_N[k], \;\;\; k=0,...,N-1 \\ \end{align}\,\! $


$ \text{(c)} \quad y[n]=\left\{\begin{array}{ll}x[n/2], & n \text{ is even},\\ 0, & n \text{ is odd},\end{array} \right. n=0,...,2N-1 $

$ \begin{align} {\color{White}abcde} Y_{2N}[k] &= \sum_{n=0}^{2N-1}y[n]e^{-j\frac{2\pi}{2N}kn} \\ &= \sum_{n=0, n\text{ even}}^{2N-1}x\left[\frac{n}{2}\right]e^{-j\frac{2\pi}{2N}kn}, \;\;\; \text{Let } n=2m \\ &= \sum_{m=0}^{N-1}x[m]e^{-j\frac{2\pi}{2N}k2m} \\ &= \sum_{m=0}^{N-1}x[m]e^{-j\frac{2\pi}{N}km}, \;\;\; k=0,...,2N-1 \\ &= X_N[k], \;\;\; k=0,...,2N-1 \\ \end{align}\,\! $



$ \text{(d)} \quad y[n]=x[2n], \;\;\; n=0,...,\frac{N}{2}-1, \;\; N\text{ even.} $

$ \begin{align} {\color{White}abcde} Y_{\frac{N}{2}}[k] &= \sum_{n=0}^{\frac{N}{2}-1}x[2n]e^{-j\frac{2\pi}{N/2}kn} = \sum_{n=0, n\text{ even}}^{N-1}x[n]e^{-j\frac{2\pi}{N}kn} \\ &= \sum_{n=0}^{N-1} \frac{1}{2}\left(1+(-1)^n\right)x[n]e^{-j\frac{2\pi}{N}kn} \\ &= \frac{1}{2}\sum_{n=0}^{N-1} x[n]e^{-j\frac{2\pi}{N}kn} + \frac{1}{2}\sum_{n=0}^{N-1} (-1)^n x[n]e^{-j\frac{2\pi}{N}kn} \\ &= \frac{1}{2}\sum_{n=0}^{N-1} x[n]e^{-j\frac{2\pi}{N}kn} + \frac{1}{2}\sum_{n=0}^{N-1} e^{-j\frac{2\pi}{N}\left(\frac{N}{2}\right)n} x[n]e^{-j\frac{2\pi}{N}kn}, \;\;\; (e^{j\pi n}=(-1)^n) \\ &= \frac{1}{2}X_N[k] + \frac{1}{2}X_N\left[k-\frac{N}{2}\right], \;\;\; k=0,...,\frac{N}{2}-1 \\ \end{align}\,\! $


Credit: Prof. Charles Bouman

Back to Lab Week 7 Quiz Pool

Back to ECE 438 Fall 2010 Lab Wiki Page

Back to ECE 438 Fall 2010

Alumni Liaison

Prof. Math. Ohio State and Associate Dean
Outstanding Alumnus Purdue Math 2008

Jeff McNeal