Digital Signal Processing

1
Digital Signal Processing

• Modal analysis and testing
• S. Ziaei Rad

2
Fourier Analysis

• Fourier series
• Fourier Transform
• Discrete Fourier series

3
Fourier series
Assume that x(t) is a periodic function in time.
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Fourier series(Alternative form)
A-
B-
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Fourier Transform
A non-periodic function x(t) which satisfies the
condition
Can be represented by
where
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Fourier Transform(Alternative form)
And
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Discrete Fourier Transform (DFT)
A function which is defined only at N discrete
points can be represented by a finite series.
where
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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.

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DFTExample
Let N10 In time domain we have
In frequency domain, we have
or
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Discrete Fourier Transform (DFT)
x(t) IS PERIODIC IN (FINITE) TIME T ALSO IS
DEFINED ONLY AT N DISCRETE POINTS

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DFT Spectrum
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DFT Spectrum
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DFT Spectrum
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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.
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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
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DFT
The basic equation that is solved to determine
spectral composition is

or
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FFT

• 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

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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
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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.

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Leakage
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.

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.

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Windowing
Hanning
Exponential
Boxcar
Cosine-taper
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Windowing
Boxcar
Hanning
Cosine-taper
Exponential
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Windowing
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Effect of Hanning Window on Discrete Fourier
Transform
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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

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Filtering
Narrow-band
High-pass
Notch
Band-limited
Frequency and time domain characteristics of
common filters
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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

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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.

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.

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
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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
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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

35
Different interpretations of multi-sample
averaging
Sequential
Overlap