Frequency-Domain Analysis - Introduction to Digital Signal Processing and Filter Design - page 160

( we will use these two equations for the DFT-IDFT pair in the remaining pages ):

N − 1 N − 1

1
N

1
N

X(k)e j( 2 π/N)kn X(k)W − kn , x(n) = = 0 ≤ n ≤ N − 1 (3.79)

k = 0 k = 0

N − 1 N − 1

x(n)e − j( 2 π/N)kn x(n)W kn ,

X(k) = = 0 ≤ k ≤ N − 1 (3.80)

n = 0 n = 0

In Figure 3.29b, we have shown the DFT as a subset of the Fourier series
coefficients X p (k) ,for k − 1 ) . But we can choose any other

N consecutive samples as the DFT of x(n) [e.g.,

= 0 , 1 , 2 ,...,(N

− [ (N − 1 )/ 2] ≤ n ≤ [ (N −

1 )/ 2]], so we will use the notation N = n modulo N to denote that n ranges

over one period of N samples.
Given a nonperiodic discrete-time function x(n) , we constructed a mathe-
matical artifact x p (n) and derived the Fourier series representation for it and
also derived its inverse to get x p (n) . Then we defined the DFT and IDFT as
argued above so that we could determine the frequency response of the non-
periodic function as samples of the DTFT X(e jω ) at N equally spaced points
ω k = ( 2 π/N)k . We know x(n) is nonperiodic, but since X(e jω ) is periodic with

aperiod2 π , X(e jω k ) = X p (k) is periodic with a period N , so one can choose

the range n in Equations (3.79) and (3.80).

The two equations for the DFT and IDFT give us a numerical algorithm
to obtain the frequency response at least at the N discrete frequencies, and
by choosing a large value for N , we get a fairly good idea of the frequency
response for x(n) . 6 Indeed, we show below that from the samples of X(k) ,we
can reconstruct the DTFT of x(n)

= N

X(e jω ) , which is a function of the contin-
uous variable ω . This is the counterpart of Shannon's reconstruction formula to
obtain x(t) from its samples x(n) , provided x(n) is bandlimited and the sam-
pling period T s <(π/ω b ) . There are similar conditions to be satisfied in deriving
the formula in the frequency domain, to reconstruct X(e jω ) from its samples
X(e jω k )

=

= X(k) .

3.6.4 Reconstruction of DTFT from DFT
First we consider the DTFT of x(n) and substitute x(n) by the formula (3.79)
for finding the IDFT of X(k) as explained below:

1
N

X(k)e j( 2 πkn/N) e − jωn

N − 1 N − 1 N − 1

X(e jω ) x(n)e − jωn = =

n = 0 n = 0 k = 0

N − 1 N − 1

1
N

e j( 2 πkn/N) e − jωn

= X(k) (3.81)

k = 0 n = 0

6 Later we will discuss what is known as the “picket fence effect,” because of which we may not get
a fairly good idea of the frequency response.