Introduction to Digital Signal and System Analysis - Tutorsindia

Introduction to Digital Signal and System Analysis
Spectral Analysis by DFT
7.6 Applications of digital spectral analysis
- Detection of harmonics of interest from noisy background
A sinusoidal signal may be mixed with strong noise and hard to be observed. It is appropriate to use the Fourier spectrum
to detect the sinusoidal signal from the noise. In the frequency domain, a signal may be displayed and extracted easily
from noise background. In the following Figure 7.6, the upper one is the signal mixed with noise in the time domain. It
is hard to observe what harmonic components are contained inside the signal. The middle curve is the Fourier spectrum
where a strong harmonic appears. Therefore, it is easy to remove the recognisable noise by reducing its noise spectrum
and to restore the harmonic shown in the lower plot using the inverse DFT.
- Fault detection
The distribution of spectra often characterises the operating condition of mechanical systems, and sudden rising of some
components or any other changes in it may indicate an occurrence of fault. Key machinery or even the whole production
lines can be monitored and early faults can be reported to prevent from unscheduled shutdown, which causes losses or
even disasters. Vibration or acoustic signals are normally picked up and digital spectra are calculated regularly with an
appropriate interval. A computer is programmed to make comparison with the historical data to report any abnormality.
- Identification of unknown systems
An unknown system is being deliberately excited by a suitable known signal – often an impulse or a step signal – and its
response is being measured, then yielding the frequency properties of the system. See Figure 7.7.
In the time domain, the input-output relationship of a system is in a form of convolution:
y[
n]
=
N
∑ − 1
k = 0
x[
k]
h[
h − k]
(7.10)
Note that this is a periodic or circular convolution, different from Eq.(3.16), as the signal x [n]
is assumed periodic. Take
the DFT for both sides, using the convolution property of DFT:
Y [ k]
= H[
k]
X [ k]
(7.11)
Therefore, the frequency response function of the system can be identified by
H [ k]
=
Y[
k]
X [ k]
(7.12)
In practice, due to noise in measurement, the frequency response function must be obtained by averaging of multiple
records of the input and output.
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