1. Digital Signal Processing
Modal analysis and testing
S. Ziaei Rad

2. Fourier Analysis Fourier series Fourier Transform
Discrete Fourier series
S. Ziaei-Rad

3. Fourier series Assume that x(t) is a periodic function in time.
S. Ziaei-Rad

4. Fourier series (Alternative form)B-
S. Ziaei-Rad

5. Fourier Transform
A non-periodic function x(t) which satisfies the condition:
Can be represented by:
where
S. Ziaei-Rad

6. Fourier Transform (Alternative form)
And
S. Ziaei-Rad

7. Discrete Fourier Transform (DFT)
A function which is defined only at N discrete points can be represented by a finite series.
where
S. Ziaei-Rad

8. Discrete Fourier Transform Alternative form
Note that:
This form of DFT is most commonly used on digital spectrum analyser.
The DFT assumes that the function x(t) is periodic.
It is important to realize that in the DFT, there are just a discrete number
of items of data in either form, i.e. N values for x and N values for Fourier series.
S. Ziaei-Rad

9. DFT Example Let N=10 In time domain we have:
In frequency domain, we have: or
S. Ziaei-Rad

10. Discrete Fourier Transform (DFT)
x(t) IS PERIODIC IN (FINITE) TIME T ALSO IS DEFINED ONLY AT N DISCRETE POINTS
WHERE
Etc. *
S. Ziaei-Rad

11. DFT Spectrum
S. Ziaei-Rad

12. DFT Spectrum
S. Ziaei-Rad

13. DFT Spectrum
S. Ziaei-Rad

14. DFT 1- The input signal is digitized by an A-D converter.
2- The input is recorded as a set of N discrete values, evenly
spaced in period T.
3-The sample is periodic in time T.
4- There is a relation between the sample length T, the number
of discrete values N, the sampling (or digitizing) rate
and the range of resolution of the frequency spectrum, i.e. .
is called the Nyquist frequency.
S. Ziaei-Rad

15. DFT
1- Usually, the size of transform (N) is fixed for an analyser, therefore
the frequency range and frequency resolution is only determined by
the length of the sample.
2- The basic equation (*) will be used to determine the coefficient
S. Ziaei-Rad

16. DFT
The basic equation that is solved to determine spectral composition is: ** or
S. Ziaei-Rad

17. FFT - Leakage - Averaging - Windowing - Filtering
Some efforts has been devoted to equation (**) for calculation
of spectral coefficients.
Cooley and Tukey (1960) introduced an algorithm called “Fast
Fourier Transform (FFT)”. The method requires N to be an integral
power of 2 and the values usually taken is between 256 to 4096.
There are number of features of digital Fourier analysis, if not
properly treated, can give rise to erroneous results. These are
generally the result of discretisation approximation and the limited
length of the time history.
- Aliasing Zooming
- Leakage Averaging
- Windowing Filtering
S. Ziaei-Rad

18. Aliasing
- Aliasing is a problem from discretisation of the originally continuous
time history.
- With this descretisation process, the existence of very high frequency
in the original signal may well be misinterpreted if the sampling rate
is too slow.
The phenomenon of aliasing
a- Low-frequency signal
b- High-frequency signal
S. Ziaei-Rad

19. Leakage
- Leakage is a direct consequence of a finite length of time history.
In Fig. a, the signal is perfectly periodic in the time window T
and the spectrum is a single line.
In Fig. b, the periodicity assumption is not valid and the spectrum
is not at a single frequency.
Energy has ‘leaked’ into a number of spectral lines in close to the
true frequency.
Leakage is a serious problem in many application of DSP, including
FRF measurements.
S. Ziaei-Rad

20. Leakage
S. Ziaei-Rad

21. Leakage Ways of avoiding leakage:
Changing the duration of the measurement sample length to
match any underlying periodicity in the signal. However, it is
difficult to determine the period of signal.
Increasing the duration of the measurement period, T, so that
the separation between spectral lines is finer.
Adding zeros to the end of the measured sample (zero padding)
In this way we partially achieving the preceding result but without
requiring more data.
Or by modifying the signal sample obtained in such a way to
reduce the severity of leakage. The process is called ‘windowing’.
S. Ziaei-Rad

22. Windowing
- In many situation, the most practical solution to the leakage problem
is windowing.
- There are a range of different windows for different classes of
problems.
Windowing is applied to the time signal before performing the
Fourier Transform.
x(t) measured signal
w(t) window profile
Or in Frequency Domain
Where * denotes the convolution process.
S. Ziaei-Rad

23. Windowing
Hanning
Exponential
Boxcar
Cosine-taper
S. Ziaei-Rad

24. Windowing
Boxcar
Hanning
Cosine-taper
Exponential
S. Ziaei-Rad

25. Windowing
S. Ziaei-Rad

26. Effect of Hanning Window on Discrete Fourier Transform
S. Ziaei-Rad

27. Filtering
-This is another signal conditioning process which has a direct parallel
with windowing.
In filtering, we simply multiply the original signal spectrum by the
frequency characteristic of the filter.
or in time domain
Common types of filters are:
High pass
Low pass
Narrow-band
Notch
S. Ziaei-Rad

28. Filtering Frequency and time domain characteristics of common filters
Narrow-band
High-pass
Notch
Band-limited
Frequency and time domain characteristics of common filters
S. Ziaei-Rad

29. Improving resolution
Inadequate frequency resolution especially at lower end of the
frequency range and for lightly-damped systems occurs because of
-limited number of discrete points available
- maximum frequency range to be covered
- the length of time sample
Possible actions to improve the resolution
Increasing transform size
Zero padding
Zoom
S. Ziaei-Rad

30. Increasing transform size
An immediate solution to this problem would be to use larger
transform. This gives finer resolution around the region of
interest. This caries the penalty of providing more information
than required.
Until recently, the time and storage requirements to perform the
DFT were a limiting factor.
- Transform size of order 2000 to 8000 are standard.
S. Ziaei-Rad

31. Zero padding
To maintain the same overall frequency range, but to increase
resolution by n, a signal sample of n times the duration is needed.
One way is to add a series of zeros to the short sample of actual
signal to create a new sample which is longer than the original
measurement and thus provide the desire finer resolution.
The fact is that no additional have been provided while apparently
greater detail in the spectrum is achieved. It is not a genuinely finer
spectrum, rather it is the coarser spectrum that interpolated and
smoothed by the extension of the analysed record.
An example of the effect and potential dangers of zero padding is
shown in the next slide.
S. Ziaei-Rad

32. Zero padding Results using zero padding to improve resolution.
a- DFT of data between 0 to T1
b- DFT of data padded to T2
c- DFT of full record 0 to T2
S. Ziaei-Rad

33. Zoom
- The common solution to the need for finer frequency resolution is
to zoom on the frequency range of interest and to concentrate all
the lines into a narrow band between
There are different ways of doing this but perhaps the easiest one is
to use a frequency shifting process coupled with a controlled aliasing
device.
Spectrum of signal
Band-pass filter
S. Ziaei-Rad

34. Averaging
When analyzing random vibrations signal, we obtain estimates for
the spectral densities and correlation functions which are used to
characterize this type of signal.
Generally, it is necessary to perform an averaging process, involving
several individual time records, before a confident results is obtained.
Two major considerations which determine the number of average:
- the statistical reliability
- the removal of random noise from the signals
S. Ziaei-Rad

35. Different interpretations of multi-sample averaging
Sequential
Overlap
S. Ziaei-Rad