Digital Signal Processing Reference
In-Depth Information
1
1
0
0
−1
−1
0
5
0
20
40
60
80
(a) Sample
(b) Sample
1
4
1
2
0
0
0
−2
−1
−1
−4
0
10
0
10
0
10
(c) Sample
(d) Sample
(e) Sample
5
0
−5
0
10
20
30
40
50
60
70
80
(f) Sample
Figure 3.29:
(a) Eight-sample impulse response; (b) Chirp, with third subsequence of eight-samples
marked; (c) Eight-sample impulse response padded with another eight zeros; (d) Third eight-sample
subsequence from (b), padded with zeros; (e) First 15 samples of circular convolution of (c) and (d),
performed using DFTs; (f ) Result from (e), after being added to previous result, giving a valid linear
convolution up to sample 24.
3.17
DF T LEAKAGE
3.17.1 ON-BIN/OFF-BIN: DF T LEAKAGE
Consider as an example a 16 sample sequence. The test correlator frequencies that the DFT will use are
-7:1:8. Any signal frequency equal to one of these will have a high correlation at the same test correlator
frequency, and a zero-valued correlation at all other test correlator frequencies due to orthogonality. Such
integer-valued signal frequencies are described as
On-Bin
or evoking an on-bin response, which is a
response confined to a single bin.
In general, noninteger signal frequencies (i.e., not equal to any of the test frequencies 0,1,2, etc)
will evoke some response in most bins. This property is usually referred to as
Leakage
, i.e.,
Off-Bin
signal energy “leaks” from the closest DFT bin into other bins.
Example 3.32.
Demonstrate DFT Leakage by taking the DFT of two sequences, each of which is 64
samples long, the first of which contains a five-cycle cosine, and the second of which contains a 5
.
3-cycle
cosine.
Figure 3.30, in plot (a), shows the DFT of the signal containing the five-cycle signal; note that
only Bins
±
5 are nonzero (Bin -5 appears as Bin 59 since the DFT uses
k
= 0:1:N-1).
