Discrete and Digital Signals and Systems - Digital Signal Processing: An Introduction with MATLAB and Applications - page 98

w r ( n )

1

Rectangular window
( N = 11 )

Time, t = n T s (sec)
T s = 0.2 sec

0 1 2 3 4 5 10

n , Sample count

| W r ( f ) |

11

f , Hz

0

−f s −f s /2 f s /2 f s = 5

Fig. 2.23 The rectangular time window and its magnitude spectrum (with total length N = 11
points, mid-point M = (N - 1)/2 = 5 points, and sampling frequency f s = 5 Hz)

Gibbs Phenomenon

The Gibbs phenomenon pertains to the oscillatory behavior in the frequency
response corresponding to a truncated digital impulse response sequence. In
the previous subsection the approximated impulse response was obtained by
truncating a shifted version of the ideal low-pass impulse response (h 1 (n) =
h L (n - M)) to get the truncated impulse response h Lt (n).This truncation process is
equivalent to multiplying h 1 (n) in the time domain by a rectangular time window
w r (n) = P N (n - M). This time window has a Fourier transform of the form

sinc ð Nf = f s Þ

sinc ð f = f s Þ e jM2pf = f s ;

W r ð f Þ¼ N

which is a sinc-like frequency function with a major lobe and many side lobes as
seen in Fig. ( 2.23 ). This multiplication in the time domain is equivalent to con-
volving H 1 (f) with W r (f) in the frequency domain. This convolution with a function
having substantial side-lobes is the reason for the oscillations which appear in the
magnitude response.
To reduce the effect of the oscillations in H Lt (f), one can multiply the ideal
impulse response with a non-rectangular window which has lower amplitude side-
lobes. Some suitable windows include the Hamming window, the Hanning win-
dow, the Blackman window, the Gaussian window and the Kaiser window [ 1 ]If
one does use this kind of smooth windowing one typically needs a longer impulse
response to have an equivalent quality of approximation.