Digital Signal Processing Reference
In-Depth Information
1.5
1
0.5
0
0
5
10
15
20
25
30
(a) n
1.5
1
0.5
0
0
5
10
15
20
25
30
(b) Bin
1.5
1
0.5
0
0
5
10
15
20
25
30
(c) Bin
Figure 3.17:
(a) A 32-point Unit Impulse sequence synthesized from cosines of length 32 and frequencies
of 0 to 31; (b) Real part of DFT of same; (c) Imaginary part of DFT of same.
is 0.5 Hz, which is the reciprocal of the total sampling duration. We'll use the term
Bin width
in this
topic to mean frequency resolution or bin spacing relative to the sampled signal's actual (or original)
frequencies.
Figure 3.18 shows this concept in generic form for a total sampling duration of
T
seconds. We see
at (a) the signal, at (b) the cosine and sine correlators for or Bin 0, at (c), the cosine and sine correlators for
Bin 1, at (d) for Bin 2, and at (e), the correlators for Bin 3. These sets of correlators have frequencies over
the sequence length
N
that go all the way up to
N/
2 for
N
even or
(N
1
)/
2 for
N
odd. The equivalent
signal frequency of Bin 1 is the reciprocal of
T
, and all higher bins represent harmonics thereof. Thus,
the equivalent bin (signal) center frequencies are 0
,
1
/T
, 2
/T
, 3
/T
, ...
L/T
with
L
=
N/
2 for
N
even or
(N
−
−
1
)/
2 for
N
odd.
Example 3.17.
A signal is sampled for 2 seconds at a sampling rate of 8 kHz, resulting in 16,000
samples. Give the equivalent signal (center) frequency of Bin 1, and give the highest bin number and its
equivalent signal frequency.
The center frequency of Bin 1 in signal terms is 1
/T
= 0.5 Hz. The highest bin index is 16000/2
= 8000. The equivalent signal frequency would be 8000/T = 4000 Hz. This is consistent with the sample
rate of 8 kHz, which can only represent nonaliased frequencies up to its Nyquist rate of 4 kHz.
3.13 BIN WIDTH AND SAMPLE RATE
Note that bin width or frequency resolution is completely independent of sample rate. The maximum
nonaliased passband or bandwidth is one-half the sample rate, which is independent of the total sampling
duration.
