Page 319 - Schaum's Outline of Theory and Problems of Signals and Systems

306 FOURIER ANALYSIS OF DISCRETE-TIME SIGNALS AND SYSTEMS [CHAP. 6

C. Relationship between the DFT and the Fourier Transform:

By definition (6.27) the Fourier transform of x[n] defined by Eq. (6.91) can be
expressed as
N- 1
X(fl) = x[n] e-j"" (6.97)
n = 0
Comparing Eq. (6.97) with Eq. (6.92), we see that

Thus, X[k] corresponds to the sampled X(fl) at the uniformly spaced frequencies
fl = k27r/N for integer k.

D. Properties of the Dm

Because of the relationship (6.98) between the DFT and the Fourier transform, we
would expect their properties to be quite similar, except that the DFT X[k] is a function
of a discrete variable while the Fourier transform X(R) is a function of a continuous
variable. Note that the DFT variables n and k must be restricted to the range 0 I n,
k < N, the DFT shifts x[n - no] or X[k - k,] imply x[n -no],,, or X[k - k,],,, .,
where the modulo notation [m],,, means that

for some integer i such that

0 [mImod~ <N (6.100)
For example, if x[n] = 6[n - 31, then
x[n - 4],,, = 6[n - 7],,,, = S[n - 7 + 61 = 6[n - 11
The DFT shift is also known as a circular shift. Basic properties of the DFT are the
following:

2. Time ShifCing:

3. Frequency Shifiing:

4. Conjugation:

'*In] ~~*[-~lmodN
where * denotes the complex conjugate.