Digital Signal Processing Reference
In-Depth Information
n
N
−
1
N
−
1
N
−
1
n
n
−
j
2
Π
k
∑
∑
∑
H
=
h
e
N
=
−
h
cos(
2
Π
k
)
−
j
h
sin(
2
Π
k
);
n
k
k
k
N
N
k
=
0
k
=
0
k
=
0
n
N
∑
−
=
1
1
j
2
Π
k
h
=
H
e
N
;
k
n
N
n
0
The frequency band thus no longer has an infinitely fine resolution and
is described only at discrete frequency interpolation points. The band ex-
tends from DC to half the sampling frequency and then continues symmet-
rically or point-to-point-symmetrically up to the sampling frequency. The
real-time graph is symmetrical up to half the sampling frequency and the
imaginary part is point-to-point-symmetrical. The frequency resolution is a
function of the number of points in the window of observation and on the
sampling frequency.
f
1
∆
f
=
s
;
∆
t
=
;
The following applies:
N
f
s
The Discrete Fourier Transform (DFT), in reality, actually corresponds
to a Fourier analysis within the observed time window of the band-limited
signal. It is thus assumed that the signal in the observed time window con-
tinues periodically. This assumption results in “uncertainties” in the analy-
sis so that the Discrete Fourier Transform can only supply approximate in-
formation about the actual frequency band. 'Approximate' in as much as
the areas preceding and following the time window are not taken into con-
sideration and the signal window is sharply truncated. However, the DFT
can be solved by simple mathematical and numerical means and it func-
tions both forwards and in the reverse direction in the time domain (In-
verse Discrete Fourier Transform - IDFT, Fig. 6.6.). The result of per-
forming a DFT on a real time domain signal interval is a discrete complex
spectrum (real and imaginary parts). The IDFT transforms the complex
spectrum back into a real time domain signal again. In reality, however,
the section of time domain signal cut out and transformed into the fre-
quency domain has been converted into a periodic signal.
Once a rectangular time domain signal segment has been windowed, the
spectrum corresponds to a convolution of a sin(x)/x function with the
original spectrum of the signal. This produces different effects which in a
spectrum analysis done by means of the DFT disturb and affect the meas-
urement result to a greater or lesser extent. In test applications, therefore,
the choice would be not to select a rectangular window function but, e.g.
cos
2
function which would cut out a smoother window and lead to fewer
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