Fourier Transform and Signal Spectrum - Digital Signal Processing: A Practitioner's Approach - page 84

20

1

16

15

0.5

3

29

0 10

−0.5 5

−1

0

0 10 20 30 40

0 10 20 30 40

Samples

Frequency bin
number

(a) (b)

Figure 4.6 Windowing effect

mathematically, the DFT of this sequence must result in a unit sample at f .
Numerically, let us consider f ¼ 1 10 implying that we have chosen 10 samples/

cycle, as shown in Figure 4.6(a). We have limited our observation to 32 samples.
Moving from 3 Figure 4.6(a) to Figure 4.6(b) can be depicted using (4.11):

=

Fig 4 6 ð b Þ

| {z }
W 32

: 6 ð a Þ! Fig 4 :

:

ð Þx!S

Note that Figure 4.6(b) corresponds to j S j .
Ideally, at bins 3 and 29 we should have obtained a value of 32 2 ¼ 16 and the

rest of the places should have been zero, but in reality we obtained something
different, as shown in Figure 4.6(b). This spectral distortion is explained in
Section 1.4. It is attributed to the fact that a rectangular window has the form
sin ð x x and its associated leakage in the sidelobes. To mitigate the problem of limited
observations, we multiply the observations point by point with another sequence w
as in Figure 4.7(a). Operationally we write it as

=

Fig 4 7 ð a Þ

| {z }
w :ð x!z

: 6 ð a Þ! Fig 4 :

:

:ðÞ denotes point-by-point multiplication of two sequences. The
non-rectangular windowed spectrum is shown in Figure 4.7(b). This meets one of
the criteria that the value of the spectrum be zero other than at f in an approximate
sense.

The operation

3

Actually we are moving from the time domain to the frequency domain.