Fourier Theory

7/17/2019 Fourier Theory

http://slidepdf.com/reader/full/fourier-theory-568ee781391c4 1/131

Fourier Theory: Overview

• spectral analysis = stationary processes + Fourier

• basic idea behind Fourier theory: given either

− real/complex-valued function g(·) defined over R or

−real/complex-valued sequence {g

t : t∈Z}

want to write (represent, synthesize) g(·) or {gt} as

“f ≥0

”A(f )cos(2πf t) + B(f )sin(2πf t) = “

f

”C (f )e−i2πft

where eix ≡ cos(x) + i sin(x) and i = √ −1

SAPA2e–51 III–1



Four Flavors of Fourier Theory: I

1. g(·) periodic with period T

− “

f

” is sum over f n = n/T , n ∈ Z

−continuous time/discrete frequency

2. g(·) square integrable:

∞−∞

|g(t)|2 dt < ∞

− “

f

” is integral over (−∞,∞) (i.e., R)

− continuous time/continuous frequency

III–2



Four Flavors of Fourier Theory: II

3. {gt} square summable:∞

t=−∞|gt|

2 < ∞

− “

f

” is integral over [−1/2, 1/2]

− discrete time/continuous frequency

4. {gt : t = 0, 1, . . . , N − 1}, a finite sequence

− “f

” is sum over f n = n/N , n = 0, 1, . . . , N − 1

− discrete time/discrete frequency

• all used in spectral analysis!

• task: define C (f ) for each flavor (known as Fourier coefficients)

III–3



Continuous Time/Discrete Frequency: I

• assume g p(·) is periodic with period T , i.e., g p(t + T ) = g p(t)for all t, and square integrable over one period: T/2

−T/2|g p(t)|2 dt < ∞

• example with period T = 1

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SAPA2e–53 III–4



Continuous Time/Discrete Frequency: II

• definitions

− nth Fourier coefficient, n ∈ Z:

Gn

≡

1

T

T/2

−T/2

g p(t)e−i2πf nt dt, f n

≡

n

T interpretation of Gn: covariance between g p(·) and complexexponential (if similar, |Gn| large)

−mth order Fourier approximation:

g p,m(t) ≡m

n=−m

Gnei2πf nt

(least squares approximation – see pp. 55–6)

SAPA2e–53 III–5



Continuous Time/Discrete Frequency: III

• can show:

limm→∞

T/2

−T/2

g p(t)− g p,m(t)2 dt = 0

• shorthand for above:

g p(t) ms

=∞

n=−∞Gnei2πf nt,

where right-hand side is Fourier series representation of g p(·)

• note: mean-square equality is not the same as pointwise equal-ity (see pp. 56–7)

• notation:g p(·) ←→ {Gn}

SAPA2e–53 III–6



Continuous Time/Discrete Frequency: IV

• to see that the definition for Gn makes sense, suppose

g p(t) =∞

n=−∞C nei2πf nt

for some set of constants C n

• multiply both sides of the above by e−i2πf mt:

g p(t)e−i2πf mt =∞

n=−∞C nei2π(f n−f m)t

• integrate both sides with respect to t over [−T /2, T /2]: T/2

−T /2g p(t)e−i2πf mt dt =

∞n=−∞

C n

T /2

−T/2ei2π(f n−f m)t dt

SAPA2e–53 III–7



Continuous Time/Discrete Frequency: V

• make use of fact that T/2

−T/2ei2π(f n−f m)t dt =

T/2

−T/2ei2π(n−m)t/T dt =

T, if n = m;

0, if n = m

to show that T/2

−T/2g p(t)e−i2πf mt dt =

∞n=−∞

C n

T/2

−T/2ei2π(f n−f m)t dt = T C m,

i.e.,

C m = 1T

T/2

−T/2g p(t)e−i2πf mt dt = Gm,

the mth Fourier coefficient for this very special g p(·)

SAPA2e–53 III–8



Continuous Time/Discrete Frequency: VI

• Parseval’s theorem: T/2

−T/2

g p(t)

2

dt = T ∞

n=−∞|Gn|

2

• proof: T/2

−T /2

g p(t)2 dt =

T /2

−T/2

n

Gnei2πf nt

m

G∗me−i2πf mt

dt

=

n

m

GnG∗m T/2

−T/2 ei2π(f n

−f m)t

dt

=

n

GnG∗nT = T

n

|Gn|2

SAPA2e–53, 54 III–9



Continuous Time/Discrete Frequency: VII

• can regard left-hand side of Parseval as “energy” in g p(·) over[−T /2, T /2] T/2

−T/2 g p(t)

2

dt = T ∞

n=−∞|Gn|

2

• corollary (why?): T/2

−T/2

g p(t)− g p,m(t)2 dt = T

|n|>m

|Gn|2

• corollary:1

T

T /2

−T/2

g p(t)2 dt =

∞n=−∞

|Gn|2

left-hand side is “power” in g p(·) (related to variance)




Continuous Time/Discrete Frequency: VIII

• Q: what are energy & power over [−mT /2,mT/2]?

• can decompose power into pieces associated with f n’s

• define discrete power spectrum for g p(·): S n ≡ |Gn|2

• Q: can we recover g p(·) from S n?

SAPA2e–75 III–11



Continuous Time/Discrete Frequency: IX

• as example, consider 2π periodic function of Equation (54c):

g p(t) ≡ 1− φ2

1 + φ2 − 2φ cos(t),

which is square integrable when |φ| < 1 with Gn = φ|n|

(theabove is related to the spectrum for an AR(1) process)

• mth order Fourier approximation:

g p,m(t) =m

n=−m

Gnei2πf nt = 1 + 2m

n=1

φn cos(nt)

• following plots show g p(·) and its Fourier approximation g p,m(·)of orders m = 4, 8, 16 and 32 when φ = 0.9




4th order Fourier Approximation g p,4(·) of g p(·)

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32nd order Fourier Approximation g p,32(·) of g p(·)

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Discrete Power Spectrum {S n} for g p(·)

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III–17



{S n} on a Decibel Scale (i.e, 10log10(S n))

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Continuous Time/Continuous Frequency: I

• assumption

− g(·) square integrable:

∞−∞

|g(t)|2 dt < ∞

• definition of Fourier transform (or analysis) of g(·):

G(f ) ≡ ∞−∞

g(t)e−i2πf t dt, −∞ < f < ∞

• can recover g(·) from G(·) (Fourier synthesis):

g(t) ms

= ∞−∞G(f )e

i2πf tdf

g(·) is inverse Fourier transform of G(·)

• can motivate above using g p,m(·)




Truncation and Periodic Extension of Function g(·)

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Continuous Time/Continuous Frequency: II

• let gT (t) be periodic wth period T such that gT (t) = g(t) fort ∈ [−T /2, T /2]

• using continuous time/discrete frequency theory, we have

g(t) ≡ gT (t) ms=

∞n=−∞

T/2

−T/2g(u)e−i2πf nu du

ei2πf nt

1

T

≈ ∞−∞

∞−∞

g(u)e−i2πfu du

ei2πft df

for large T , where the integral within the parentheses is theFourier transform of g(·)




Continuous Time/Continuous Frequency: III

• shorthand notation g(·) ←→ G(·) indicates that−G(·) is Fourier transform of g(·)

− g(·) is inverse Fourier transform of G(·)

− g(·) & G(·) form a Fourier transform pair

• many other conventions for defining Fourier transform exist!• Parseval’s theorem: ∞

−∞|g(t)|2 dt =

∞−∞

|G(f )|2 df

left-hand side is “energy” in g(·) (Q: what would “power” be?)• define energy spectral density function: |G(f )|2

• can write G(f ) = |G(f )|eiθ(f ), where |G(f )| is the amplitudespectrum and θ(f ) is the phase function




Continuous Time/Continuous Frequency: IV

• example: e−πt2 ←→ e−πf 2

− change of variable yields gσ(·) ←→ Gσ(·), where

gσ(t) = 1

(2πσ2

)

1/2e−t2/(2σ2), Gσ(f ) = e−2π2f 2σ2

− gσ(·) is probability density function for Gaussian (normal)random variable with mean zero and variance σ2 (standarddeviation σ)

−note: Gσ(·) real-valued so can see |Gσ(·)|2 easily




gσ(·) and Gσ(·) for σ = 1

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Time-/Band-Limited Functions

• g(·) time-limited to [−T, T ] if g(t) = 0 for all |t| > T − lots of examples!

• g(·) band-limited to [−W, W ] if G(f ) = 0 for all |f | > W

−male speech limited to 8000 Hz (= cycles/second)

− orchestra limited to 20,000 Hz− has representation

g(t) ms

=

W

−W

G(f )ei2πf t df

− can be diff erentiated arbitrary number of times (very “smooth”)• Q: can g(·) be both time- and band-limited?

• of considerable interest (Chapters 6 and 8): time-limited se-quences that are close to band-limited




Reciprocity Relationships: I

• g(·) ←→ G(·) implies that, in a mean-square sense, G(·) uniquelydetermines g(·) and vice versa

• can regard g(·) and G(·) as two representations for a singlemathematical entity, per this quote from Bracewell (2000):

“We may think of functions and their transforms as oc-cupying two domains, sometimes referred to as the upperand the lower, as if functions circulated at ground leveland their transforms in the underworld (Doetsch, 1943).There is a certain convenience in picturing a function as

accompanied by a counterpart in another domain, a kindof shadow which is associated uniquely with the functionthrough the Fourier transformation, and which changes asthe function changes.”




Reciprocity Relationships: II

• can quantify how changing a function in the time-domain af-fects its frequency-domain representation via three reciprocityrelationships

1. similarity theorem

2. equivalent width3. fundamental uncertainty relationship

SAPA2e–63, 64, 65 III–29



Similarity Theorem

• g(·) ←→ G(·) implies |a|1/2g(at) ←→ G(f /a)/|a|1/2

• for a > 1, |a|1/2g(at) formed by

− contracting g(·) horizontally

−expanding g(·) vertically

whereas G(f /a)/|a|1/2 formed by

− expanding G(·) horizontally

− contracting G(·) vertically

• as an example, considerg(t) =

1

(2π)1/2e−t2/2 ←→ G(f ) = e−2π2f 2

for a = 1, 2 and 4




|a|1/2g(at) and G(f /a)/|a|1/2 for a = 1

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Equivalent Width: I

• measures concentration of signal in time

• sensible when g(·) is real, nonnegative, even & continuous at 0

• definition:

widthe {g(·)} ≡ ∞

−∞ g(t) dt

g(0)

• width of rectangular signal whose

− height is g(0)

− area is the same as the area under curve of g(·)




Equivalent Width of g(t) = exp(−|t|)

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Equivalent Width: II

• note: area = ∞−∞

g(t) dt = G(0) & g(0) = ∞−∞

G(f ) df

• implies

widthe {g(·)} = G(0) ∞

−∞G(f ) df =

1

widthe {G(·)}• product of widths of signal & transform = unity




Fundamental Uncertainty Relationship: I

• if g(·) real & nonnegative with unit area, then g(·) is probabilitydensity function (PDF)

• consider a uniform PDF r(·; µr, W r) that is centered at µr andhas width 2W r and height 1/2W r

• variance σ2r is related to spread of a PDF:

σ2r ≡

∞−∞

(t− µr)2 r(t; µr, W r) dt = W 2r

3

• relationship between “natural width” 2W r

and standard devi-ation σr is thus

2W r = 2σr√

3




Fundamental Uncertainty Relationship: II

• if g(·) has nonunit area, form g(t) (same width!):

g(t) ≡ g(t)/

∞−∞

g(t) dt

• variance width of g(·) is widthv {g(·)} ≡ 2σg√

3

• for general g(·), use widthv|g(·)|2

• suppose |g(·)|2 integrates to unity (i.e., is a PDF):

∞

−∞

|g(t)|2 dt = 1 = ∞

−∞

|G(f )|2 df

• let σ2g and σ2

G be variances of |g(·)|2 and |G(·)|2

• can show (pp. 66–7) that σ2g × σ2

G ≥ 1/16π2, with equalityholding only in the Gaussian case

SAPA2e–65, 66, 67 III–38



Convolution Theorem: I

• briefly: convolution in time domainequivalent to multiplication in frequency domain

• convolution of g(·) & h(·) is this function of t:

∞−∞ g(u)h(t− u) du ≡ g ∗ h(t)

− assumes integral exists

− “reflect and translate” second function, i.e., h(·)

−g

∗h(·) notation for function defined above

− change of variable shows h ∗ g(·) same as g ∗ h(·)




Illustration of Convolution: I

• consider convolution of

g(u) =

3/4, |u| < 3/4 and

0, otherwiseand h(u) =

1, |u| < 1/4 and

0, otherwise.

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• following plots show g(u), h(t− u), g(u) · h(t− u) and g∗h(t)




Illustration of Convolution: II

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Illustration of Convolution: III

• now consider convolution of rectangles of equal width:

g(u) =

3/4, |u| < 1/2 and

0, otherwiseand h(u) =

1, |u| < 1/2 and

0, otherwise.

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• following plots show g(u), h(t− u), g(u) · h(t− u) and g∗h(t)




Illustration of Convolution: IV

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Convolution Theorem: II

• Fourier transform of g ∗ h(·) is the product of the individualtransforms for g(·) and h(·): ∞

−∞g ∗ h(t)e−i2πft dt = G(f )H (f )

(this is Exercise [73])

• thus g ∗ h(·) ←→ G(·)H (·)

• variety of convolution theorems in literature that stipulate con-ditions for above to hold




Convolution as Smoothing Operation: I

• regard h(·) as a signal, g(·) as a smoother (filter) and g ∗ h(·)as smoothed version of h(·)

• example

− a signal: h(t) =

L

l=1 Al cos (2

πf lt +

φl)

− a smoother: g(t) = 1

(2πσ2)1/2e−t2/(2σ2), where we can re-

gard σ as an adjustable smoothing parameter

− smoothed version of h(·) (Exercise [75]):

g ∗ h(t) =L

l=1

e−(σ2πf l)2/2Al cos (2πf lt + φl)




Convolution as Smoothing Operation: II

• let’s compare h(·) and its smoothed version:

h(t) =L

l=1

Al cos (2πf lt + φl) , g∗h(t) =L

l=1

e−(σ2πf l)2/2Al cos (2πf lt + φl)

• note that− frequencies f l and phases φl are unchanged

− 0 < e−(σ2πf l)2/2 < 1 is an attenuation factor

−smoother shrinks amplitudes toward 0

− as f l → 0, attenuation factor increases to 1

− as f l →∞, attenuation factor decreases to 0

− reduces amplitudes of high frequency terms




Specific Examples of Smoothing: I

• consider example with f 1 = 1/6 and f 2 = 3, namely,

h(t) = 5 cos (2π1

6t + 0.5) + cos (2π3t + 1.1)

along with three settings for σ (0.1, 0.25 and 0.625)




g(·), h(·) and g ∗ h(·) for σ = 0.1

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g(·), h(·) and g ∗ h(·) for σ = 0.25

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g(·), h(·) and g ∗ h(·) for σ = 0.625

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Specific Examples of Smoothing: II

• attenuation factors:f 1 = 1/6 f 2 = 3

σ = 0.1 0.99 0.17σ = 0.25 0.97 0.0

σ = 0.625 0.81 0.0




Specific Examples of Smoothing: III

• another smoother:

r(t) =

1/2δ , −δ ≤ t ≤ δ ;

0, otherwise.

• smoothed version of h(·) (note: sinc(u)

≡sin(πu)/(πu)):

r∗h(t) = 1

2δ

t+δ

t−δ h(u) du =

Ll=1

sinc (2f lδ )Al cos (2πf lt + φl)

• sinc (2f lδ ) varies about 0 (not monotonic in f l)

• consider same example as before

h(t) = 5 cos (2π1

6t + 0.5) + cos (2π3t + 1.1)

along with three settings for δ (1/8, 1/6 and 1/4)

SAPA2e–75, 76, 77 III–52



r(·), h(·) and r ∗ h(·) for δ = 1/8

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III–53



r(·), h(·) and r ∗ h(·) for δ = 1/6

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r(·), h(·) and r ∗ h(·) for δ = 1/4

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Specific Examples of Smoothing: IV

• δ = 1/8 damps down f 2 term• δ = 1/6 eliminates f 2 term completely

• δ = 1/4 causes ripples to appear!

• will prefer smoothers with monotonic attenuation




Cross and Autocorrelations

• variations on convolution idea• cross-correlation of g(·) and h(·):

g h∗(t) ≡ ∞

−∞

g(u + t)h∗(u) du

• can show (Exercise [3.13a]): g h∗(·) ←→ G(·)H ∗(·)

• letting h(·) = g(·) yields autocorrelation:

g g∗(t)

≡ ∞

−∞g(u + t)g∗(u) du

• have g g∗(·) ←→ G(·)G∗(·) = |G(·)|2




Autocorrelation Width

• leads to another measure of width:

widtha {g(·)} ≡ widthe {g g∗(·)}

=

∞−∞ g g∗(t) dt

g g∗(0)

[78f]=

∞−∞ g(t) dt2 ∞

−∞ |g(t)|2 dt

(will prove useful in Chapters 6, 7 and 8)




Comparison of Three Measures of Width: I

• plot shows rectangular PDF g(·) along with− equivalent width: widthe {g(·)} (red lines)

− variance width: widthv {g(·)} (red lines)

−autocorrelation width: widtha {g(·)} (red lines)

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Comparison of Three Measures of Width: II

• plot shows standard Gaussian PDF g(·) along with− equivalent width: widthe {g(·)} (dashed lines)

− variance width: widthv {g(·)} (blue lines)

−autocorrelation width: widtha {g(·)} (red lines)

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Discrete Time/Continuous Frequency: I

• assumptions about g(·)− has finite energy

− is continuous at t∆t, t ∈ Z, where ∆t > 0

• extract samples gt

≡g(t∆t), where ∆t is time interval between

samples (i.e., sampling interval)

• need to assume sequence {gt} is square summable:∞

t=−∞

|gt|2 <

∞




{gt} Obtained from Samples of g(·) with ∆t = 1

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Discrete Time/Continuous Frequency: II

• definition: discrete Fourier transform of {gt} is

G p(f ) ≡ ∆t

∞t=−∞

gte−i2πft∆t

− first motivation: if g(·) ←→ G(·), then

G(f ) ≡ ∞−∞

g(t)e−i2πft dt

≈ ∆t

∞

t=−∞

g(t∆t)e−i2πft∆t = G p(f )

− second motivation: use Dirac delta functions (p. 80)

III–63



Discrete Time/Continuous Frequency: III

• reason for p subscript (note e−i2πt = 1 for integer t):

G p(f + 1∆t

) = ∆t

∞t=−∞

gte−i2π(f + 1

∆t)t∆t

= ∆t ∞t=−∞

gte−i2πft∆te−i2πt = G p(f )

G p(·) is periodic with period T = 1/∆t (deja vu!)

III–64



Discrete Time/Continuous Frequency: IV

• apply continuous time/discrete frequency theory to G p(·)• Fourier coefficients for G p(·) are, say,

gn ≡ 1

T T /2

−T /2

G p(t)e−i2πf ntdt with f n = n

T = n ∆t

= ∆t

1/2∆t

−1/2∆t

G p(t)e−i2πtn∆t dt

• Fourier synthesis of G p(·) is thus

G p(t) =∞

n=−∞gnei2πf nt =

∞n=−∞

gnei2πtn∆t




Discrete Time/Continuous Frequency: V

• changing (i) t to f and (ii) n to t yields

gt = ∆t

1/2∆t

−1/2∆t

G p(f )e−i2πf t∆t df

G p(f ) =∞

t=−∞

gtei2πf t∆t

• letting gt = g−t∆t yields

gt = 1/2∆t

−1/2∆t

G p(f )ei2πft∆t df

G p(f ) = ∆t

∞t=−∞

gte−i2πf t∆t

2nd equation is definition; 1st gives inverse DFT




Discrete Time/Continuous Frequency: VI

• notation: {gt}←→ G p(·)• Parseval etc. falls out readily

• two questions of interest

1. given just g−

m, . . . , gm (a finite sample), how well can G p(·)be approximated?

2. how are G(·) and G p(·) related?

• answers to questions involve discussion of

− leakage, convergence factors (windows)− aliasing




Finite Sample Approximation of G p(·): I

• assuming ∆t = 1, can approximate G p(·) using

G p,m(f ) ≡m

t=−m

gte−i2πf t

[82]= (2m + 1)

1/2

−1/2

G p(f )D2m+1(f

−f ) df ,

where D2m+1(·) is Dirichlet’s kernel, which is proportional tothe Fourier transform of a “rectangular” sequence {rt}

• can regardm

t=−m

gte−i2πft = (2m + 1) 1/2

−1/2G p(f )D2m+1(f − f ) df

as example of “inverse” convolution theorem:

{gt × rt}←→ (2m + 1)G p ∗D2m+1(·)




Finite Sample Approximation of G p(·): II

• given g−m, . . . , gm, can argue that G p,m(·) is best approxima-tion to G p(·) in least squares sense

• consider plots of D2m+1(·) for m = 4, 16 and 64




D2m+1(·) for m = 4

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D2m+1(·) for m = 16

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D2m+1(·) for m = 64

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Finite Sample Approximation of G p(·): III

• as m →∞, (2m +1)D2m+1(·) converges to a Dirac δ function,which is good in view of

G p,m(f ) = (2m + 1)

1/2

−1/2G p(f )D2m+1(f − f ) df

• in contrast to a δ function, the approximation has

1. central lobe of finite width

2. sidelobes (some of which are negative)

which, as the following figures illustrate, can

1. smear out features, resulting in a loss of resolution (attribut-able to having just a subsequence from {gt})

2. cause leakage and a “Gibbs” phenomenon

SAPA2e–82, 83, 84 III–73



{gt} with “Twin Peaks” DFT G p(·)

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III–74



“Twin Peaks” G p(·) (DFT of {gt})

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“Twin Peaks” G p(·) & Approximation G p,4(·)

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{gt} with Rectangular DFT G p(·)

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III–79



Rectangular G p(·) (DFT of {gt})

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Rectangular G p(·) & Approximation G p,4(·)

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Cesaro Sums: I

• can reduce leakage & Gibbs using Cesaro sums• let . . . , u−2, u−1, u0, u1, u2, . . . be an infinite sequence

• form mth partial sum: sm ≡m

t=−

m

ut

• form average of partial sums of orders 0, . . . , m− 1:

am ≡ 1

m

m−1 j=0

s j =m

t=−m

1− |t|

m

ut

(to see this, work out what am is for, e.g., m = 3)

• above called (two-sided) Cesaro sum

• theorem: if sm → s, then am → s also




Cesaro Sums: II

• application: let

sm =m

t=−m

gte−i2πft = G p,m(f )

• since G p,m(f ) → G p(f ), must also have

G(C )

p,m(f ) ≡m

t=−m

1− |t|

m

gte−i2πf t → G p(f )

i.e., G

(C )

p,m(·) is another approximation for G p(·)




Cesaro Sums: III

• claim:G

(C ) p,m(f ) ≡

mt=−m

1− |t|

m

gte−i2πf t

[86]= m 1/2

−1/2G p(f

)D2

m(f −

f ) df

• sketch of proof:

− (1− |t|/m) ∝ convolution of {rt} with itself

−FT of {rt}

∝Dm(·)

− thus FT of (1− |t|/m) ∝ D2m(·)

− FT of (1− |t|/m)× gt = convolution of FTs

• D2m(·) related to Fejer’s kernel (Chapter 6)




D2m+1(·) and D2m(·) for m = 4

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D2m+1(·) and D2m(·) for m = 16

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D2m+1(·) and D2m(·) for m = 64

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Cesaro Sums: IV

• as plots of D2m+1(·) and D2m(·) show,

−D2m(·) has smaller sidelobes (hurray!)

−D2m(·) has nonnegative sidelobes (hurray!)

−D2m(·) has wider central lobe (boo!)

• following plots illustrate tradeoff s between approximations basedon D2m+1(·) and D2

m(·)




G p(·) Approximates Based on D2m+1(·) & D2m(·), m = 4

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C ` S



Cesaro Sums: V

• more generally, can approximate G p(·) usingm

t=−m

ctgte−i2πf t,

where {ct}’s are called convergence factors or a window


G ( ) G( )



Relating G p(·) to G(·) – Aliasing: I

• assume general ∆t, i.e., ∆

t is not necessarily unity

• {gt}←→ G p(·) implies gt =

1/(2∆t)

−1/(2∆t)G p(f )ei2πft∆t df

• g(·) ←→ G(·) and gt = g(t∆t) imply

gt = ∞−∞G(f )ei2πf t∆t df

=∞

k=−∞

(2k+1)/(2∆t)

(2k−1)/(2∆t)G(f )ei2πf t∆t df

= ∞k=−∞

1/(2∆t)

−1/(2∆t)G(f + k/∆t)ei2π(f +k/∆t)t∆t df,

where f ≡ f − k/∆t

SAPA2e–86, 87, 88 III–95

R l i G ( ) G( ) Ali i II



Relating G p(·) to G(·) – Aliasing: II

• since ei2π(f +k/∆t)t∆t = ei2πft∆tei2πkt = ei2πft∆t, we have

gt =∞

k=−∞

1/(2∆t)

−1/(2∆t)G(f + k/∆t)ei2π(f +k/∆t)t∆t df

= 1/(2∆t)

−1/(2∆t) ∞

k=−∞G(f + k/∆

t)

ei2πft∆t

df

• since gt =

1/(2∆t)

−1/(2∆t)G p(f )ei2πft∆t df also, must have

G p(f ) =∞

k=−∞G(f + k/∆t) for |f | ≤ 1/(2∆t) ≡ f N ,

where f N is known as the Nyquist frequency


R l i G ( ) G( ) Ali i III



Relating G p(·) to G(·) – Aliasing: III

• f ± k/∆t, k = 0, are aliases of f • highest f that is not an alias of a lower frequency is f N

• following plots illustrate various aspects of aliasing


Ill t ti f Ali i (∆ 1 & f 0 5)



Illustration of Aliasing (∆t = 1 & f N = 0.5)

−"#$ −" −%#$ % %#$ " "#$

&

G(·)

G p(·)


(2 f t) (2 (f 1)t) d S l t t 0 1 8



cos(2πf t), cos(2π(f − 1)t) and Samples at t = 0, 1, . . . , 8

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R l ti G ( ) t G( ) Ali i IV



Relating G p(·) to G(·) – Aliasing: IV

• as an example, consider following function:g(t) = 2

√ πe−π

2t2/α2[cos (2πf 1t) + 0.9cos(2πf 2t)] ,

where α = 100, f 1 = 14 − 1

50 and f 2 = 14 + 1

50

• Fourier transform of g(·) is

G(f ) = e−α2(f −f 1)2 + e−α2(f +f 1)2 + 0.9e−α2(f −f 2)2 + 0.9e−α2(f +f 2)2

• both g(·) & G(·) are even functions (i.e., g(−t) = g(t) &G(−f ) = G(f )), so it suffices to look at g(t) for t ≥ 0 &G(f ) for f

≥0

• following overheads show g(·) & G(·) and gt = g(t∆t) & G p(·)for ∆t = 0.5, 1, 1.5, 2, 3, 4, 8, 16 and 32 (Nyquist frequencyindicated by vertical dashed line)


(t) d (t∆ ) ∆ 0 5



g(t) and gt = g(t∆t), ∆t = 0.5

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III–101

G(f ) d G (f ) ∆ 0 5



G(f ) and G p(f ), ∆t = 0.5

0.0 0.2 0.4 0.6 0.8 1.0

0 . 0

0 .

5

1 .

0

1 .

5

2 .

0

2 .

5

3 .

0

f

III–102

(t) d (t∆ ) ∆ 32



g(t) and gt = g(t∆t), ∆t = 32

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III–103

G(f ) and G (f ) ∆ 32



G(f ) and G p(f ), ∆t = 32

0.0 0.2 0.4 0.6 0.8 1.0

0 . 0

0 .

5

1 .

0

1 .

5

2 .

0

2 .

5

3 .

0

f

III–104

Discrete/Continuous Concentration Problem: I



Discrete/Continuous Concentration Problem: I

• assumptions (∆t = 1 for convenience)− gt real-valued & time-limited to t = 0, . . . , N − 1

− {gt}←→ G p(·)

• energy =

N −1t=0

g2t =

1/2

−1/2G p(f )

2df

• how close can G p(·) be to bandlimited?

• for 0 < W < 1/2, consider concentration measure:

β 2(W ) ≡

W

−W

G p(f )2 df

1/2

−1/2

G p(f )2 df

SAPA2e–91, 92 III–105

Discrete/Continuous Concentration Problem: II



Discrete/Continuous Concentration Problem: II

• reduction of numerator (using |z |2

= zz ∗): W

−W

G p(f )

2

df =

W

−W

N −1t=0

gte−i2πf t

N −1t=0

gtei2πf t

df

=N −

1t=0

N −

1t=0

gtgt

W

−W ei2πf (t−t) df

=N −1

t=0

N −1

t=0

gtgtsin[2πW (t − t)]

π(t −

t) = gT Ag,

where g ≡ [g0, . . . , gN −1]T and A is an N ×N matrix whose(t, t)th element is sin [2πW (t − t)]/π(t − t)


Discrete/Continuous Concentration Problem: III



Discrete/Continuous Concentration Problem: III

• reduction of denominator (why?): 1/2

−1/2

G p(f )2 df = gT g

• hence can write concentration measure as

β 2(W ) = gT Ag/gT g

• our task is to find g that maximizes β 2(W ) for a given W


Solution to Concentration Problem: I



Solution to Concentration Problem: I

• to maximize β 2

(W ), diff

erentiate with respect to g and set tovector of zeros:dβ 2(W )

dg = 0

• says that solution g must satisfy Ag = β 2(W )g (necessary,

but not sufficient condition)• eigenvalue/eigenvector problem: Ag = λg

• solution is eigenvector, say v0(N, W ), associated with largesteigenvalue, say, λ0(N, W ) = β 2(W )

• v0(N, W ) is (subsequence) of 0th order discrete prolate spher-oidal sequence (DPSS) – also called Slepian sequence

• A is positive definite, so all N eigenvalues are positive

SAPA2e–94, 95 III–108

Solution to Concentration Problem: II



Solution to Concentration Problem: II

• can show that eigenvalues are distinct, so order as:0 < λN −1(N, W ) < · · · < λ1(N, W ) < λ0(N, W ) < 1

• Q: why must λ0(N, W ) be less than unity?

• first 2NW (Shannon number) eigenvalues close to 1, after whichλk(N, W )’s fall off rapidly to 0

• can use eigenvectors to form orthonormal basis:

v j(N, W )T vk(N, W ) = 1, j = k;

0, otherwise.• following plots give some examples

SAPA2e–94, 95 III–109

v0(N W ) with N = 32 and W = 4/N = 1/8



v0(N, W ) with N = 32 and W = 4/N = 1/8

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v1(N W ) with N = 32 and W = 4/N = 1/8



v1(N, W ) with N = 32 and W = 4/N = 1/8

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v2(N W ) with N = 32 and W = 4/N = 1/8



v2(N, W ) with N = 32 and W = 4/N = 1/8

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v3(N W ) with N = 32 and W = 4/N = 1/8



v3(N, W ) with N = 32 and W = 4/N = 1/8

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|G (·)|2 for v0(N W ) with N = 32 and W = 4/N = 1/8



|G p(·)| for v0(N, W ) with N = 32 and W = 4/N = 1/8

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|G (·)|2 for v1(N W ) with N = 32 and W = 4/N = 1/8




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|Gp(·)|2 for v2(N W ) with N = 32 and W = 4/N = 1/8




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|Gp(·)|2 for v3(N W ) with N = 32 and W = 4/N = 1/8



|G p( )| for v3(N, W ) with N = 32 and W = 4/N = 1/8

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v0(N, W ) with N = 99 and W = 4/N = 4/99



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v1(N, W ) with N = 99 and W = 4/N = 4/99



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|Gp(·)|2 for v0(N, W ) with N = 99 and W = 4/N = 4/99



|G p( )| 0( , W ) w 99 W / /99

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| p( )| 1( , ) / /

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| p( )| 2( , ) / /

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| p( )| 3( , ) / /

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λk(N, W ) with N = 99 and W = 4/N = 4/99 (2NW = 8)



0 20 40 60 80 100

0 .

0

0 .

2

0 .

4

0 .

6

0 .

8

1

. 0

k


Discrete Time/Discrete Frequency: I



• {gt : t = 0, . . . , N

−1}, sampled ∆t units apart

• two possible definitions for Fourier transform:

− form infinite sequence: set gt ≡ 0 for other t’s:

G p(f ) = ∆t

∞

t=−∞

gte−i2πf t∆t = ∆t

N −1

t=0

gte−i2πft∆t

useful (e.g., periodogram), but infinite number of frequencies

− define discrete Fourier transform (DFT) of {gt}:

Gn ≡

G p

(f n

) = ∆

t

N −1

t=0

gte−

i2πf nt∆t = ∆

t

N −1

t=0

gte−

i2πnt/N ,

where f n are N Fourier (standard) frequencies:

f n ≡ n/N ∆t, n = 0, 1, . . . , N − 1

SAPA2e–97, 98 III–128

Discrete Time/Discrete Frequency: II



• inverse DFT given by

gt[98]=

1

N ∆t

N −1n=0

Gnei2πnt/N

• fast Fourier transform (FFT) algorithm vs. DFT• two standard forms: ∆t = 1 or ∆t = 1/N

SAPA2e–98, 99 III–129

Discrete Time/Discrete Frequency: III



• DFT definition implies gt = 0 beyond t = 0 to N − 1• notion of zero padding

− add N −N > 0 zeros to g0, . . . , gN −1 to form g0, . . . , gN −1with gt = 0, N ≤ t ≤ N − 1

− DFT of padded sequence (f n ≡ n/N ∆t):

Gn ≡ ∆t

N −1t=0

gte−i2πnt/N = ∆t

∞t=−∞

gte−i2πnt/N = G p(f n)

evaluates G p(·) over finer grid than f n’s− useful to compute convolutions or DFT via chirp transformalgorithm (p. 101)

SAPA2e–100, 101 III–130

Discrete Time/Discrete Frequency: IV



• can also claim DFT implies periodic extension:

gt ≡ 1

N ∆t

N −1n=0

Gnei2πnt/N , t < 0 or t ≥ N ,

so {gt : t∈Z} has period N

• can use right-hand side of DFT to define Gn for all n, so

gt = 1

N ∆t

N +k−1

n=k

Gnei2πnt/N

for any integer k

• note: summary of Fourier theory, pp. 103–10

Fourier Theory

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