Signals and Linear System

1
Signals and Linear System

• Fourier Transforms
• Sampling Theorem

2
Fourier Transform

• Is the extension of Fourier series
• to non-periodic signal
• Definition of Fourier transform
• Fourier transform
• Inverse Fourier transform
• From Fourier series (T0 ? ?)

3
Properties of FT

• For a real signal x(t)
• X(f) is Hermitian Symmetry
• Magnitude spectrum is even about the origin (f0)
• Phase spectrum is odd about the origin
• f, called frequency (units of Hz), is just a
  parameter of FT that specifies what frequency we
  are interested in looking for in the x(t)
• The FT looks for frequency f in the x(t) over -?
  lt t lt ?
• F(f) can be complex even though x(t) is real
• If x(t) is real, then Hermitian symmetry

4
Properties of FT

• Linearity
• Duality
• If , Then
• Time Shift
• A shift in the time domain results in a phase
  shift in the frequency domain
• ( )

5
Properties of FT

• Scaling
• An expansion in the time domain results in a
  contraction in the frequency domain, and vice
  versa

6
Properties of FT

• Modulation
• Multiplication by an exponential in the time
  domain corresponds to a frequency shift in the
  frequency domain

7
Properties of FT

• Differentiation
• Differentiation in the time domain corresponds to
  multiplication by j2?f in the frequency domain

8
Properties of FT

• Convolution
• Convolution in the time domain is equivalent to
  multiplication in the frequency domain, and vice
  versa

9
Properties of FT

• Parsevals relation
• Energy can be evaluated in the frequency domain
  instead of the time domain
• Rayleighs relation

10
More on FT pairs

• See Table 1.1 at page 20
• Delta function ?? Flat
• Time / Frequency shift
• Sin / cos input
• Periodic signal ? impulses in the frequency
  domain
• sgn / unit step input
• Rectangular ?? sinc
• Lambda ?? sinc2
• Differentiation
• Pulse train with period T0
• Periodic signal ? impulses in the frequency domain

11
FT of periodic signals

• For a periodic signal with period T0
• x(t) can be expressed with FS coefficient
• Take FT
• FT of periodic signal consists of impulses at
  harmonics of the original signal

12
FS with Truncated signal

• FS coefficient can be expressed using FT
• Define truncated signal
• FT of truncated signal
• Expression of FS coefficient

13
Spectrum of the signal

• Fourier transform of the signal is called the
  Spectrum of the signal
• Generally complex
• Magnitude spectrum
• Phase spectrum
• Illustrative problem 1.5
• Time shifted signal
• Same magnitude, Different phase
• Try it by yourself with Matlab !

14
Sampling Theorem

• Basis for the relation between continuous-time
  signal and discrete-time signals
• A bandlimited signal can be completely described
  in terms of its sample values taken at intervals
  Ts as long as Ts ? 1/(2W)

1
15
Impulse sampling

• Sampled waveform
• Take FT

16
Reconstruction of signal

• Low pass filter
• With Bandwidth 1/(2Ts) and Gain of Ts

17
Reconstruction from sampled signal

• If we have sampled values
• x(-2Ts), x(-Ts), 0, x(Ts), x(2Ts)
• With Nyquist interval (or Nyquist rate)
• Ts 1/(2W)
• Then we can reconstruct x(t) using
• Example Figure 1.17 at page 24

18
Aliasing or Spectral folding

• If sampling rate is Ts gt 1/(2W)
• Spectrum is overlapped
• We can not reconstruct original signal with
  under-sampled values
• Anti-aliasing methods are needed

19
Discrete Fourier Transform

• DFT of discrete time sequence xn
• Relation between FT and DFT
• FFT(Fast Fourier Transform)
• Efficient numerical method to compute DFT
• See fft.m and fftseq.m, for more information
• Example Try Illustrative problem 1.6 by yourself

20
DFT in Matlab

• fft.m and ifft.m with finite samples
• Definition of DFT
• Definition of IDFT
• Time and frequency is not appeared explicitly
• Just definition implemented on a computer to
  compute N values for the DFT and IDFT
• N is chosen to be N2m
• Zero padding is used if N is not power of 2

21
FFT in matlab

• A sequence of length N2m of samples of x(t)
  taken at Ts
• Ts satisfies Nyquist condition
• Ts is called time resolution
• FFT gives a sequence of length N of sampled Xd(f)
  in the frequency interval 0, fs1/Ts
• The samples are apart
• is called frequency resolution
• Frequency resolution is improved by increasing N

22
Some remarks on short signal

• FT works on signals of infinite duration
• But, we only measure the signal for a short time
• FFT works as if the data is periodic all the time

23
Some remarks on short signal

• Sometimes this is correct

24
Some remarks on short signal

• Sometimes wrong

25
Frequency leakage

• If the period exactly fits the measurement time,
  the frequency spectrum is correct
• If not, frequency spectrum is incorrect
• It is broadened

26
Frequency domain analysis of LTI system

• The output of LTI system
• Take FT (using convolution theorem)
• Where the Transfer Function of the system
• The relation between input-output spectra

27
Homeworks

• Illustrative problem 1.7
• Problems
• 1.10, 1.12, 1.14, 1.15

28
More on Sampling

• Most real signals are analog
• The analog signal has to be converted to digital
• Information lost during this procedure
  (Quantization Error)
• Inaccuracies in measurement
• Uncertainty in timing
• Limits on the duration of the measurement

29
More on Sampling

• Continuous analog signal has to be held before it
  can be sampled

30
More on Sampling

• The sampling take place at equal interval of time
  after the hold
• Need fast ADC
• Need fast hold circuit
• Signal is not changing during the time the
  circuit is acquiring the signal value
• Unless, ADC has all the time that the signal is
  held to make its conversion
• We dont know what we dont measure

31
More on Sampling

• In the process of measuring signal, some
  information is lost

32
Aliasing

• We only sample the signal at intervals
• We dont know what happens between the samples

33
Aliasing

• We must sample fast enough to see the most rapid
  changes in the signal
• This is Sampling theorem
• If we do not sample fast enough
• Some higher frequencies can be incorrectly
  interpreted as lower ones

34
Aliasing

• Called aliasing because one frequency looks
  like another

35
Aliasing

• Nyquist frequency
• We must sample faster than twice the frequency of
  the highest frequency component

36
Antialiasing

• We simply filter out all the high frequency
  components before sampling
• Antialias filters must be analog
• It is too lte once you have done the sampling

37
More on sampling Theorem

• The sampling theorem does not say the samples
  will look like the signal

38
More on Sampling Theorem

• Sampling theorem says there is enough information
  to reconstruct the signal
• Correct reconstruction is not just draw straight
  lines between samples

39
More on Sampling Theorem

• The impulse response of the reconstruction filter
  has sinc (sinx/x) shape
• The input to the filter is the series of discrete
  impulses which are samples
• Every time an impulse hits the filter, we get
  ringing
• Superposition of all these rings reconstruct the
  proper signal

40
Frequency resolution

• We only sample the signal for a certain time
• We must sample for at least one complete cycle of
  the lowest frequency we want to resolve

41
Quantization

• When the signal is converted to digital form
• Precision is limited by the number of bits
  available

42
Uncertainty in the clock

• Uncertainty in the clock timing leads to errors
  in the sampled signal

43
Digitization errors

• The errors introduced by digitization are both
  nonlinear and signal dependent
• Nonlinear
• We can not calculate their effect using normal
  maths.
• Signal dependent
• The errors are coherent and so cannot be reduced
  by simple means

44
Digitization errors

• The effect of quantiztion error is often similar
  to an injection of random noise