Frequency domain Analysis of LTI systems

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Frequency domain Analysis of LTI systems

• Summarized by
• Neetesh Purohit
• Lecturer, IIIT,
• Allahabad, UP, India
• http//profile.iiita.ac.in/np/

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Contents

• Discrete-time Fourier Series (DTFS)
• Discrete-time Fourier Transform (DTFT)
• Frequency Domain Characteristics of DTLTI systems
• DTLTI systems as frequency selective filters
• Inverse systems and deconvolution

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A little history

• Astronomic predictions by Babylonians/Egyptians
  likely via trigonometric sums.

• 1669 Newton stumbles upon light spectra (specter
  ghost) but fails to recognise frequency
  concept (corpuscular theory of light, no waves).

• 18th century two outstanding problems
• celestial bodies orbits Lagrange, Euler
  Clairaut approximate observation data with linear
  combination of periodic functions
  Clairaut,1754(!) first DFT formula.
• vibrating strings Euler describes vibrating
  string motion by sinusoids (wave equation). BUT
  peers consensus is that sum of sinusoids only
  represents smooth curves. Big blow to utility of
  such sums for all but Fourier ...

• 1807 Fourier presents his work on heat
  conduction ? Fourier analysis born.
• Diffusion equation ? series (infinite) of sines
  cosines. Strong criticism by peers blocks
  publication. Work published, 1822 (Theorie
  Analytique de la chaleur).

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• 19th / 20th century two paths for Fourier
  analysis - Continuous Discrete.

• CONTINUOUS
• Fourier extends the analysis to arbitrary
  function (Fourier Transform).
• Dirichlet, Poisson, Riemann, Lebesgue address
  FS convergence.
• Other FT variants born from varied needs (ex.
  Short Time FT - speech analysis).

• DISCRETE Fast calculation methods (FFT)
• 1805 - Gauss, first usage of FFT (manuscript in
  Latin went unnoticed!!! Published 1866).
• 1965 - IBMs Cooley Tukey rediscover FFT
  algorithm (An algorithm for the machine
  calculation of complex Fourier series).
• Other DFT variants for different applications
  (ex. Warped DFT - filter design signal
  compression).
• FFT algorithm refined modified for most
  computer platforms.

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Fourier analysis - tools
Input Time Signal Frequency spectrum
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FS synthesis
Square wave reconstruction from spectral terms
Convergence may be slow (1/k) - ideally need
infinite terms. Practically, series truncated
when remainder below computer tolerance (?
error). BUT Gibbs Phenomenon.
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From FS to FT
Frequency spacing ?0 !
Note ak?2 a0 as k?0 ? 2 a0 is plotted at k0
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FT - example
FT of 2?-wide square window
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Discrete Fourier Series (DFS)
DFS generate periodic ck with same signal period
analysis
synthesis
N consecutive samples of sn completely describe
s in time or frequency domains.
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DFS of periodic discrete 1-Volt square-wave
Discrete signals ? periodic spectra. Compare to
continuous rectangular function
Ck is a ratio of sinc(x) functions. It has
maximum value at x0 and zero value at x /- (p,
2p, 3p .)
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Difference Between CTFS and DTFS

• Fourier series of continuous signals can extends
  from 8 to 8, as there is no limit on frequency
  of analog signals.
• A discrete time signal of fundamental period N
  can consists of frequency components separated by
  2p/N radians or f1/N cycles.
• Thus its Fourier series can contain at most N
  frequency components and will be periodic of same
  period.
• Example 4.2.1, 4.2.2

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Discrete Time FT (DTFT)
1. X(f) exists if xn is absolutely summable
i.e. ?xnlt8 2. continuous frequency domain!
3. The range of interest is p ltwlt p as
X(f2p) X(f).
DTFT defined as
ESD Sxx(w) X(w)2 Ex 1/2p ?(-p to p) Sxx(w)
dw
Problems 4.2.3, 4.2.4
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• X(w) X(z) at z 1 thus X(w) does not exists
  (in strict sense) if ROC of X(z) does not include
  the unit circle.
• If frequency response function H(w) exists then
  the system will be stable.
• Using concept of impulse the Fourier Transform
  representation can be extended to some sequences
  which are neither absolutely nor square summable.
  e. g. Unit step function.

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• u(n) starts at n0 and any signal which have an
  abrupt jump contains infinite frequency
  components (definitely within 0 to p for discrete
  signal). (prove it?) example 4.2.5
• At w0, U(w0) can be approximated as an impulse
  of weight p and for other values it can be
  evaluated from expression of U(w).

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Frequency Response Function of DTLTI systems

• H(w) It is the Fourier Transform of h(n). (find
  an expression for y(n) in terms of H(w) for a
  complex exponential I/P x(n) Aejwn)
• Eigenfunction of a system is an I/P signal that
  produces an O/P that differs from I/P by a
  constant multiplicative factor. (there may be
  more than one eigenfunctions to a system)
• The above multiplicative factor is called an
  eigenvalue of the system. (example 5.1.1)

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• H(w) HR(w) j HI(w) H(w)ejF(w)
• H(w) ?(k -8 to 8) h(k)e-jwk
  
• ?(k -8 to 8) h(k).cos(wk) - j ?(k -8
  to 8) h(k).sin(wk)
• (compare above expressions)
• Magnitude response H(w) sqrt HR2(w)
  HI2(w) is an even function thus symmetric about
  w0.
• Phase response F(w) tan-1HI(w)/HR(w) is an odd
  function thus antisymmetric about w0.
• Thus, if we have evaluated the above values for 0
  to p, we can know their values for p to 0.(ex.
  5.1.2, 5.1.3)

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Transient and steady state response

• y(n) an1y(-1)?(k0 to n) akx(n-k) for 1st
  order
• If x(n) Aejwn then,
• y(n) an1y(-1) A?(k0 to n) (ae-jw)k ejwn
• The term in is a finite GP of n1 terms
• y(n) an1y(-1) Aan1e-jw(n1)ejwn/(1-ae-jw)
  Aejwn/(1-ae-jw)
• The system is stable if alt1 and as n?8 the
  first two terms vanish thus sum of first two
  terms represents transient response and the last
  term represents steady state response,
• yss(n) Aejwn/(1-ae-jw) AH(w) ejwn.

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ESD of O/P

• Y(w)2 H(w)2X(w)2
• Syy(w) H(w)2Sxx(w)
• As per Weiner-Kinchiene theorem Fourier Transform
  of autocorrelation function of a sequence is
  equal to ESD i.e. Sxx(w) Frxx(l)
• Examples 5.1.5, 5.2.1

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Working of DTLTI systems as Filters

• Y(w)H(w)X(w)
• H(w) acts as a weighting function or spectral
  shaping function to the different frequency
  components in the I/P signal.
• Every LTI system works as a filter even though it
  may not necessarily completely block/pass any or
  all frequency component.

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• Distortionless Transmission
• Attenuation and delay is not considered as signal
  distortion
• Constant magnitude and negative linear phase is
  required
• First derivative of phase response is called
  envelope delay/group delay and it should be a
  constant.
• Ideal filters are unrealizable
• Rectangular shape of H(w) in frequency domain
  requires sync(n) shape of h(n) in time domain,
  which is a noncausal sequence hence practically
  it can not be implemented

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Pole-Zero placement method for filter designing

• Locate the poles (zeros) near points of the unit
  circle corresponding to frequencies to be
  emphasized (deemphasized) w varies from 0 to
  p in z-plane.
• All poles should be inside the unit circle
  however zeros can be placed any where.
• All complex poles or zeros must occur in
  complex-conjugate pairs in order for filter
  coefficients to be real.
• Gain of system (b0) should be selected such that
  H(w0)1 where is the w0 central frequency of
  pass band. (example 5.3.1 and 5.3.2)

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LP to HP filter transformation

• H(w) is periodic (period 2p) and symmetric
  function i.e. H(w) H(-w)
• If H(w) represents a LPF then H(w-p) will
  represent a HPF and vice versa i.e.
• Hhp(w) Hlp(w-p)
• Thus, hhp(n) (ejp)nhlp(n) (-1)nhlp(n)
• Above transformation can be done by changing the
  sign of all odd-numbered samples of h(n).
• It is equivalent of changing sign of coefficients
  of odd-numbered ak and bk in its difference
  equation. (prove it?)

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Some important filters

• Digital Resonators
• Notch Filters
• Comb Filter
• All Pass Filters

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Inverse systems

• If a system produces the O/P y(n) in response to
  an I/P x(n) then its inverse system will produce
  x(n) as O/P on applying y(n) as I/P. Thus,
  cascading a system with its inverse system
• H(z).HI(z) 1 or h(n)hI(n) ?(n)
• Zeros of a system becomes poles of its inverse
  system and vice versa. (inverse system of an FIR
  system is an all pole system) (example 5.4.1,
  5.4.2)

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• Invertibility A system is said to be invertible
  if there is one-to-one correspondence between its
  I/P and O/P (monotonic relationship) e.g. y(n)
  ax(n) or x(n-k) but y(n) x2(n) is a
  non-invertible system.
• The equalizer is an inverse system to the
  communication channel (treated as the system)
  which is used to cancel the distortion introduced
  by the channel.
• Ideal channel equalizers are not possible due to
  noninvertible property of channels. But it can be
  approximated.

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Minimum-phase, maximum-phase and mixed-phase
systems

• Consider two simple FIR systems with reciprocal
  zeros e.g. H1(z) 1(1/2)z-1 and H2(z)
  (1/2)z-1
• In this case, if zeros of a system falls inside
  the unit circle the zeros of other system will
  definitely fall outside the unit circle.
• Such systems have exactly similar magnitude
  response but the difference is reflected in phase
  response

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• When w is varied from 0 to p, the phase
  difference F(wp) F(w0) of the system,
  having all zeros inside unit circle, is minimum
  and vice versa.
• Same phase characteristics are also observed in
  IIR systems thus DTLTI systems can be classified
  on the basis of these phase characteristics.

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• Minimum-phase system
• all zeros of H(z) must be inside the unit
  circle.
• Maximum-phase system
• all zeros of H(z) must be outside the unit
  circle.
• Mixed phase systems
• some zeros are inside and some zeros are outside
  the unit circle.

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• Causal inverse system of a minimum phase system
  will always be stable (Why?) and vice versa.
• Inverse systems of maximum and mixed phase
  systems will always be unstable.
• Example 5.4.4

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System Identification and deconvolution

• The process of determining the characteristics
  h(n) or H(w) of an unknown system by a set of
  measurements performed on the system is called
  system identification.
• Deconvolution is an operation which is used in
  system identification. The inverse system
  operation that takes y(n) as I/P and produces
  x(n) is called deconvolution.
• Various methods are available to solve the
  deconvolution problem.

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• First find Y(z) from observed O/P y(n) and X(z)
  from known x(n) then H(z) Y(z)/X(z) and
  h(n)Z-1H(z).
• (it is suitable if y(n) is a finite length
  sequence)
• Second as we know,
• y(n) ?(k0 to n) h(k)x(n-k)
• thus h(0)y(0)/x(0) and
• h(n) y(n) - ?(k0 to n-1) h(k)x(n-k)/x(0)
• (practically we are required to stop at some
  point, thus there will be some error if h(n) is
  an infinite duration sequence)

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• Third we know that,
• ryx(l) y(l)x(-l) h(l)x(l)x(-l) h(l)rxx(l)
• It can be solved for h(n) by recursive method
  (second approach).
• Taking Fourier transform, Syx(w) H(w)Sxx(w)
• or H(w) Syx(w)/Sxx(w)
• If an I/P is selected such that its ESD is an
  unit sample sequence then H(w)Syx(w) or h(n)
  ryx(n).
• It is a practical and effective approach for
  system identification.