FOURIER TRANSFORMS OF DISCRETE-TIME SIGNALS - Signals, Systems, and Transforms - page 621

Frequency Resolution of the DFT

Because the DTFT from which the N frequency samples were taken is periodic with
period the discrete frequency spectrum that we compute using the DFT has a
resolution (separation between samples) of

2p,

¢Æ = 2p > N. (12.34)

This is illustrated by Figure 12.8 for where the unit circle represents one
period of the signal's discrete-time Fourier transform. From this, we see that the
choice of the number of samples of used in the calculation determines the reso-
lution of the frequency spectrum, or vice-versa; the resolution required in the fre-
quency spectrum determines the number of samples of that we must use. In the
event that a fixed number, of time-domain samples is available, but a larger num-
ber, of frequency-domain samples is required to provide adequate resolution,
zeros can be appended to the time sequence. This process is called zero
padding . We discuss applications of zero padding in Section 12.6.

N = 8, (2p)

x[n]

x[n]

N 1 ,

N 2 ,

N 2 - N 1

Im

90

1

120 60

k 6

0.8

k 7

0.6

k 5

150 30

0.4

0.2

k 0 Re

180 0

k 4

210 330

k 1

k 3

k 2

240

300

270

W k

Figure 12.8 Polar plot of .