Due 2 March 2016 at 5:00 PM
ECE 438
1.
Assignment No. 5
Spring 2016
Consider the length N signal x[n] ¹ 0, only for n = 0, 1, …, N-1. Let y[n] be a new
length M = LN signal defined by repeating x[n] L times for some integer L:
y[n] = x[nmod N], n = 0,..., LN -1.
Here the operation p mod q denotes the remainder from the division of p by q .
Find a simple expression for the M-point DFT Y[k], k = 0,…, M-1 of the signal
y[n] in terms of the N-point DFT X[l], l = 0,…, N-1, of the signal x[n].
2.
Consider the length N signal x[n] ¹ 0, only for n=0, 1, …, N-1. Let y[n] be a new
signal defined by padding x[n] with zeros to make it length M=LN for some integer
L:
ì x[n], n = 0,..., N - 1
y[n] = í
n = N,..., M - 1
î 0,
Let X(w) be the DTFT of x[n] and let Y[k], k=0,…, M-1 be the M point DFT of
y[n].
Show that Y[k] yields samples from X(w) at points w k = 2pk / M , k=0,1,…, M-1,
i.e.
æ 2pk ö÷
Y[k] = X ç
, k = 0,..., M - 1.
è Mø
3.
a.
Derive a decimation-in-time FFT algorithm for a 18-point DFT, and draw a
complete flow diagram for the algorithm.
b.
Calculate the approximate number of complex operations required to compute
the 18-point FFT and compare with the number of complex operations
required to compute the 18-point DFT directly.
4.
You have software to compute the forward DFT, but no software to compute the
inverse DFT. Devise an approach to using your forward FFT software to actually
compute the inverse DFT by first preprocessing the data, taking a forward DFT,
then performing some post-processing on the output from the FFT algorithm.
5.
You want to exactly compute an exact 640-point DFT. You have a radix 2 FFT
subroutine that computes the DFT for N = 2 M points for any integer value of M.
a.
Show how to use this subroutine to efficiently compute a 640-point DFT.
b.
Draw a block diagram for your algorithm, showing the radix 2 FFT subroutine
as a black box with no detail regarding what is inside it.
c.
Calculate the approximate number of complex operations required to compute
the 640-point DFT using your efficient approach, and compare with the
2
number of complex operations required to compute the 640-point DFT
directly.