Digital Signal Processing Reference
In-Depth Information
way, the frequency response is actually the frequency response of
an infinite long periodic signal, where the N long sequence of x
i
samples repeat over and over. Lastly, the input signal x
i
is usually
assumed to be a complex (real and quadrature) signal. The
frequency response samples X
k
are also complex. Often we are
more interested in only the magnitude of the frequency response
X
k
, which can be more easily displayed. But in order to get back
the original complex input x
i
using the IDFT, we would need the
complex sequence X
k
.
At this point, we will do a few examples, selecting N
¼
8.
For our N
¼
8 point DFT, the output gives us the distribution of
input signal energy into eight frequency bins, corresponding to
the frequencies in
Table 12.1
below. By computing the DFT
coefficients X
k
, we are performing a correlation, or trying to
match, our input signal to each of these frequencies. The
magnitude DFT output coefficients X
k
represent the degree of
match of
the time-domain signal x
i
to each frequency
component.
12.1.1 First DFT Example
Let's start with a simple time domain signal consisting of
{1,1,1,1,1,1,1,1}. Remember, the DFT assumes this signal keeps
repeating, so the frequency output will actually be that of an
indefinite string of 1s. As this signal is unchanging, then by
intuition, we will expect that zero frequency component (DC of
Table 12.1
Phase Between
Each Sample Of
Complex Exponential
Signal
6
Compute By Correlating
To Complex
Exponential Signal
k
X
k
e
0
0
X
0
for i ¼ 0,1..7
0
e
j2pi/8
for i ¼ 0,1..7
1
X
1
p
/4or45 degrees
e
j4pi/8
for i ¼ 0,1..7
2
X
2
2
p
/4or90 degrees
e
j6pi/8
for i ¼ 0,1..7
3
X
3
3
p
/4or135 degrees
e
j8pi/8
for i ¼ 0,1..7
4
X
4
4
p
/4or180 degrees
e
j10pi/8
for i ¼ 0,1..7
5
X
5
5p /4or225 degrees
e
j12pi/8
for i ¼ 0,1..7
6
X
6
6p /4or270 degrees
e
j14pi/8
for i ¼ 0,1..7
7
X
7
7p /4or315 degrees
