ELEN%20E4810:%20Digital%20Signal%20Processing%20Topic%203:%20Fourier%20domain

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ELEN E4810 Digital Signal ProcessingTopic 3
Fourier domain

1. The Fourier domain
2. Discrete-Time Fourier Transform (DTFT)
3. Discrete Fourier Transform (DFT)
4. Convolution with the DFT

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1. The Fourier Transform

• Basic observation (continuous time)A periodic
  signal can be decomposed into sinusoids at
  integer multiples of the fundamental frequency
• i.e. if
• we can approach with

Harmonics of the fundamental
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Fourier Series

• For a square wave,
• i.e.

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Fourier domain

• x is equivalently described by its Fourier
  Series parameters
• Complex form

Negative ak is equivalent to phase of p
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Fourier analysis

• How to find ck, argck?Inner product with
  complex sinusoids

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Fourier analysis

• Works if k1, k2 are positive integers,

?
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sinc

• 1 when x 0 0 when x rp, r ? 0, r 1,
  2, 3,...

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Fourier Analysis

• Thus,because real imag sinusoids in pick out
  corresponding sinusoidal components linearly
  combined in

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Fourier Transform

• Fourier series for periodic signals extends
  naturally to Fourier Transform for any (CT)
  signal (not just periodic)
• Discrete index k ? continuous freq. W

FourierTransform (FT)
Inverse FourierTransform (IFT)
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Fourier Transform

• Mapping between two continuous functions

2p ambiguity
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Fourier Transform of a sine

• Assume
• Now, since
• ...we know
• ...where d(x) is the Dirac delta function
  (continuous time) i.e.
• ? ?

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Fourier Transforms
Time Frequency
Fourier Series (FS) Continuous periodic x(t) Discrete infinite ck
Fourier Transform (FT) Continuous infinite x(t) Continuous infinite X(W)
Discrete-Time FT (DTFT) Discrete infinite xn Continuous periodic X(ejw)
Discrete FT (DFT) Discrete finite/pdc xn Discrete finite/pdc Xk


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2. Discrete Time FT (DTFT)

• FT defined for discrete sequences
• Summation (not integral)
• Discrete (normalized) frequency variable w
• Argument is ejw, not plain w

DTFT
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DTFT example

• e.g. xn anmn, a lt 1
• ?

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Periodicity of X(ejw)

• X(ejw) has periodicity 2p in w
• Phase ambiguity of ejw makes it implicit

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Inverse DTFT (IDTFT)

• Same basic form as other IFTs
• Note continuous, periodic X(ejw) discrete,
  infinite xn ...
• IDTFT is actually forward Fourier Series (except
  for sign of w)

IDTFT
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IDTFT

• Verify by substituting in DTFT

0 unlessn l
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sinc again

• Same as ?cos imag jsin part cancels

?
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DTFTs of simple sequences

• xn dn ?
• i.e. xn X(ejw)
• dn ? 1

(for all w)
xn
X(ejw)
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DTFTs of simple sequences
IDTFT

• ? over -p lt w lt p
• but X(ejw) must be periodic in w ?
• If w0 0 then xn 1 ? n
• so

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DTFTs of simple sequences

• From before
• mn tricky - not finite

( a lt 1)
DTFT of 1/2
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DTFT properties

• Linear
• Time shift
• Frequency shift

delayin frequency
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DTFT example

• xn dn anmn-1 ? ?
• dn a(an-1mn-1)
• ?

? xn anmn
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DTFT symmetry

• If xn ? X(ejw) then...
• x-n ? X(e-jw)
• xn ? X(e-jw)
• Rexn ? XCS(ejw)
• jImxn ? XCA(ejw)
• xcsn ? ReX(ejw)
• xcan ? jImX(ejw)

from summation
(e-jw) ejw
conjugate symmetry cancels Im parts on IDTFT
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DTFT of real xn

• When xn is pure real, ? X(ejw) X(e-jw)
• xcsn ? xevn xev-n ? XR(ejw)
  XR(e-jw)
• xcan ? xodn -xod-n ? XI(ejw)
  -XI(e-jw)

xn real, even ? X(ejw) even, real
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DTFT and convolution

• Convolution
• ?

Convolution becomesmultiplication

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DTFT modulation

• ModulationCould solve if gn was just
  sinusoids...

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Dual of convolution in time
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Parsevals relation

• Energy in time and frequency domains are equal
• If g h, then gg g2 energy...

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Energy density spectrum

• Energy of sequence
• By Parseval
• Define Energy Density Spectrum (EDS)

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EDS and autocorrelation

• Autocorrelation of gn
• ?
• If gn is real, G(e-jw) G(ejw), so
• Mag-sq of spectrum is DTFT of autoco

no phaseinfo.
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Convolution with DTFT

• Since
• we can calculate a convolution by
• finding DTFTs of g, h ? G, H
• multiply them GH
• IDTFT of product is result,

gn
DTFT
yn
IDTFT
hn
DTFT
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DTFT convolution example

• xn anmn ?
• hn dn - adn-1
• ?
• yn xn hn
• ?
• ? yn dn i.e. ...

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3. Discrete FT (DFT)
Discrete FT (DFT) Discrete finite/pdc xn Discrete finite/pdc Xk

• A finite or periodic sequence has only N unique
  values, xn for 0 n lt N
• Spectrum is completely defined by N distinct
  frequency samples
• Divide 0..2p into N equal steps, wk 2pk/N

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DFT and IDFT

• Uniform sampling of DTFT spectrum
• DFT
• where i.e. 1/Nth of a revolution

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IDFT

• Inverse DFT IDFT
• Check

Sum of complete setof rotated vectors 0 if l ?
n N if l n
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DFT examples

• Finite impulse
• ?
• Periodic sinusoid (r ? I)
• ?

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DFT Matrix form

• as a matrix multiply
• i.e.

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Matrix IDFT

• If
• then
• i.e. inverse DFT is also just a matrix,

1/NDN
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DFT and DTFT
continuous freq w infinite xn, -?ltnlt?
DTFT
discrete freq kNw/2p finite xn, 0nltN
DFT

• DFT samples DTFT at discrete freqs

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DFT and MATLAB

• MATLAB is concerned with sequences not continuous
  functions like X(ejw)
• Instead, we use the DFT to sample X(ejw) on an
  (arbitrarily-fine) grid
• X freqz(x,1,w) samples the DTFT of sequence x
  at angular frequencies in w
• X fft(x) calculates the N-point DFT of an
  N-point sequence x

M
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DTFT from DFT

• N-point DFT completely specifies the continuous
  DTFT of the finite sequence

periodicsinc
interpolation
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Periodic sinc

• N when Dwk 0 (-)N when Dwk/2 p 0
  when Dwk/2 rp/N, r 1, 2, ...other values
  in-between...

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Periodic sinc
Periodicsincinterpolation Xk?X(ejw)
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DFT from overlength DTFT

• If xn has more than N points, can still
  form
• IDFT of Xk will give N point
• How does relate to xn ?

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DFT from overlength DTFT
1 for n-l rN, r?I 0 otherwise
all values shifted by exact multiples of N
ptsto lie in 0 n lt N
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DFT from DTFT example

• If xn 8, 5, 4, 3, 2, 2, 1, 1 (8 point)
• We form Xk for k 0, 1, 2, 3by sampling
  X(ejw) at w 0, p/2, p, 3p/2
• IDFT of Xk gives 4 pt
• Overlap only for r -1

(N 4)
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DFT from DTFT example

• xn
• xnN (r -1)
• is the time aliased or folded down
  version of xn.

n
-1
1
2
3
4
5
6
7
8
n
-1
-2
-3
-4
-5
1
2
3
4
5
n
1
2
3
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Properties Circular time shift

• DFT properties mirror DTFT, with twists
• Time shift must stay within N-pt window
• Modulo-N indexing keeps index between 0 and N-1

0 n0 lt N
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Circular time shift

• Points shifted out to the right dont disappear
  they come in from the left
• Like a barrel shifter

delay by 2
5-pt sequence
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Circular time reversal

• Time reversal is tricky in modulo-N indexing -
  not reversing the sequence
• Zero point stays fixed remainder flips

5-pt sequence made periodic
Time-reversedperiodic sequence
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4. Convolution with the DFT

• IDTFT of product of DTFTs of two N-pt sequences
  is their 2N-1 pt convolution
• IDFT of the product of two N-pt DFTs can only
  give N points!
• Equivalent of 2N-1 pt result time aliased
• i.e.
• must be, because GkHk are exact samples of
  G(ejw)H(ejw)
• This is known as circular convolution

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Circular convolution

• Can also do entire convolution with modulo-N
  indexing
• Hence, Circular Convolution
• Written as

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Circular convolution example

• 4 pt sequences

? 1
? 2
n
? 0
? 1
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Duality

• DFT and IDFT are very similar
• both map an N-pt vector to an N-pt vector
• Duality if then
• i.e. if you treat DFT sequence as a time
  sequence, result is almost symmetric

Circulartime reversal
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DFT properties summary

• Circular convolution
• Modulation
• Duality
• Parseval

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Linear convolution w/ the DFT

• DFT ? fast circular convolution
• .. but we need linear convolution
• Circular conv. is time-aliased linear conv. can
  aliasing be avoided?
• e.g. convolving L-pt gn with M-pt hnyn
  gn hn has LM-1 nonzero pts
• Set DFT size N LM-1 ? no aliasing

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Linear convolution w/ the DFT

• Procedure (N L M - 1)
• pad L-pt gn with (at least) M-1 zeros? N-pt
  DFT Gk, k 0..N-1
• pad M-pt hn with (at least) L-1 zeros? N-pt
  DFT Hk, k 0..N-1
• Yk GkHk, k 0..N-1
• IDFTYk

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Overlap-Add convolution

• Very long gn ? break up into segments, convolve
  piecewise, overlap
• ? bound size of DFT, processing delay
• Make
• Called Overlap-Add (OLA) convolution...

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Overlap-Add convolution
hn g0n

n
hn g1n

n
hn g2n

n
valid OLA sum
hn gn

n
N
2N
3N