PRACTICAL MATLAB® FOR ENGINEERS PRACTICAL MATLAB

490 Practical MATLAB® Applications for Engineers

Let

wN = e−j(2/N)

then

Fk f n WNnk k N

n

N

()

()



0

1

∑ for 0,1,2,3, ,... −^1

R.5.90 The IDFT is given by

fn

N

Fk WNnk k N

n

N

()

() 

1 

0

1







∑

for 0,1,2,3, ,... 1

Observe that because of the exponential term e−j^2 πnk/N, f(n) becomes periodic
outside the interval 0 ≤ n ≤ N − 1.

R.5.91 Observe that DFT is related to DTFT and ZT by making the following
substitutions:
F(k) = F(e−jW), where W = 2 πk/N (DTFT)
F(k) = F(z), where z = e−j(2πk/N) (ZT)

R.5.92 DFT can be viewed as a fi nite sum representation of discrete sinusoidal of the
periodic sequence f(n), with a period or length N.

R.5.93 Observe that DFT can be viewed as a simple and practical way to approximate
DTFT by a fi nite sequence N.

R.5.94 The MATLAB command fft(fn, M)* returns the DFT of fn consisting of M elements,
where fn is given by a sequence of N elements, where M > N. In its simplest form,
fft(fn) returns the N points of DFT of fn. The magnitude and phase plots of fn can
be obtained by using the MATLAB command [fft_mag, fft_phase] = fftplot(fn, Ts),
where Ts denotes the sampling rate. The default value is Ts = 1.

R.5.95 The MATLAB function fn = ifft(F, N) returns the N points of the IDFT of F, where
the length [fn] = length [F].

R.5.96 DFT provides an effi cient approach to the numerical complexity of the evaluation
of the DTFT of a fi nite length sequence. The computational effi ciency is based on
the argument that the sequence f(n,) with length N must be an integer power of 2. If
N is not an integer power of 2, then the sequence f(n) is augmented to a length of M
by concatenating M − N zero value samples, where M is an integer power of 2. The
DFT of the new sequence f_ aug(n) with length M can now be computed effi ciently
by using the fft algorithm.

R.5.97 The DFT of an N-point real or complex discrete-time sequence is an N-complex
sequence. If the length required for the DFT is M (integer power of 2 ), where M > N,

*^ The fft algorithm was fi rst presented by J.W. Cooley and J.W. Tukey in their paper An Algorithm for the Machine
Calculations of the Complex Fourier Series, Math. Comp, April 1965.