Decimation-in-frequency FFT algorithmThe decimation-in-time FFT algorithms are all based on structuring the DFT computation by forming smaller and smaller subsequences of the input sequence x[n]. Alternatively, we can consider dividing the output sequence X[k] into smaller and smaller subsequences in the same manner.

The even-numbered frequency samples are

1,...,1,0 ][][1

0

NkWnxkXN

n

nkN

1

)2/(

)2(1)2/(

0

)2(1

0

)2( ][][][]2[N

Nn

rnN

N

n

rnN

N

n

rnN WnxWnxWnxrX

1)2/(

0

))2/((21)2/(

0

2 )]2/([][]2[N

n

NnrN

N

n

nrN WNnxWnxrX

Since

and

The above equation is the (N/2)-point DFT of the (N/2)-point sequence obtained by adding the first and the last half of the input sequence.

Adding the two halves of the input sequence represents time aliasing, consistent with the fact that in computing only the even-number frequency samples, we are sub-sampling the Fourier transform of x[n].

rnN

rNN

rnN

NnrN WWWW 22)]2/([2

2/2

NN WW

1)2/(,...,1,0 )])2/([][(]2[1)2/(

02/

NrWNnxnxrXN

n

rnN

We now consider obtaining the odd-numbered frequency points:

Since

1

)2/(

)12(1)2/(

0

)12(1

0

)12( ][][][]12[N

Nn

rnN

N

n

rnN

N

n

rnN WnxWnxWnxrX

)12(1)2/(

0

)12(1)2/(

0

)12)(2/(

1)2/(

0

)12)(2/(1

2/

)12(

)]2/([

)]2/([

)]2/([][

rnN

N

n

rnN

N

n

rNN

N

n

rNnN

N

Nn

rnN

WNnx

WNnxW

WNnxWnx

We obtain

The above equation is the (N/2)-point DFT of the sequence obtained by subtracting the second half of the input sequence from the first half and multiplying the resulting sequence by WN

n.

Let g[n] = x[n]+x[n+N/2] and h[n] = x[n]x[x+N/2], the DFT can be computed by forming the sequences g[n] and h[n], then computing h[n] WN

n, and finally computing the (N/2)-point DFTs of these two sequences.

1)2/(,...,1,0 ])2/[][(

])2/[][(]12[

2/

1)2/(

0

1)2/(

0

)12(

NrWWNnxnx

WNnxnxrX

nrN

N

n

nN

N

n

rnN

Flow graph of decimation-in-frequency decomposition of an N-point DFT (N=8).

Recursively, we can further decompose the (N/2)-point DFT into smaller substructures:

Finally, we have

Butterfly structure for decimation-in-frequency FFT algorithm:

The decimation-in-frequency FFT algorithm also has the computation complexity of O(N log2N)

Chirp Transform Algorithm (CTA)This algorithm is not optimal in minimizing any measure of computational complexity, but it has been used to compute any set of equally spaced samples of the DTFT on the unit circle.

To derive the CTA, we let x[n] denote an N-point sequence and X(ej

w) its DTFT. We consider the evaluation of M samples of X(ejw) that are equally spaced in angle on the unit cycle, at frequencies

)/2

,1,...,1,0(

0

Mw

Mk

wkwwk

When w0 =0 and M=N, we obtain the special case of DFT.

The DTFT values evaluated at wk are

with W defined as

we have

The Chirp transform represents X(ejwk) as a convolution:

To achieve this purpose, we represent nk as

1,...,1,0 ][)(1

0

MkenxeXN

n

njwjw kk

wjeW

1,...,1,0 ][)(1

0

0

MkWenxeXN

n

nknjwjwk

])()[2/1( 222 nkknnk

Then, the DTFT value evaluated at wk is

Letting

we can then write

To interpret the above equation, we obtain more familiar notation by replacing k by n and n by k:

X(ejwk) corresponds to the convolution of the sequence g[n] with the sequence Wn2/2.

1

0

2/)(2/2/ 2220][)(

N

n

nkknnjwjw WWWenxeX k

2/20][][ nnjw Wenxng

1,...,1,0 ][)(1

0

2/)(2/ 22

MkWngWeXN

n

nkkjwk

1,...,1,0 ][)(1

0

2/)(2/ 22

MnWkgWeXN

k

knnjwn

The block diagram of the chirp transform algorithm is

Since only the outputs of n=0,1,…,M1 are required, let h[n] be the following impulse response with finite length (FIR filter):

Then

otherwise

MnNWnh

n

0

1)1(][

2/2

1,...,1,0 ][][)( 2/2

MnnhngWeX njwn

The block diagram of the chirp transform algorithm for FIR is

Then the output y[n] satisfies that

Evaluating frequency responses using the procedure of chirp transform has a number of potential advantages: We do not require N=M as in the FFT algorithms, and neither N nor M need be composite numbers. => The frequency values can be evaluated in a more flexible manner. The convolution involved in the chirp transform can still be implemented efficiently using an FFT algorithm. The FFT size must be no smaller than (M+N1). It can be chosen, for example, to be an appropriate power of 2.

1,...,1,0 ][)( MnnyeX njw

In the above, the FIR filter h[n] is non-causal. For certain real-time implementation it must be modified to obtain a causal system. Since h[n] is of finite duration, this modification is easily accomplished by delaying h[n] by (N1) to obtain a causal impulse response:

and the DTFT transform values are

In hardware implementation, a fixed and pre-specified causal FIR can be implemented by certain technologies, such as charge-coupled devices (CCD) and surface acoustic wave (SAW) devices.

otherwise

NMnWnh

Nn

0

2,...,1,0][

2/)1(

1

2

1,...,1,0 ]1[)( 1 MnNnyeX njw

Two-dimensional Transform Revisited(c.f. Fundamentals of Digital Image Processing, A. K. Jain, Prentice

Hall, 1989)

One-dimensional orthogonal (unitary) transforms v=Au

u = A*T v = AH v

where A*T = A1, i.e., AAH = AHA = I. That is, the columns of AH form a set of orthonormal bases, and so are the columns of A.

The vector ak* {ak,n*, 0 n N1} are called the basis vector of A. The series coefficients v[k] give a representation of the original sequence u[k], and are useful in filtering, data compression, feature extraction, and other analysis.

10 ][][1

0,

NknuakvN

nnk

10 ][][1

0,

NnkvanuN

knk

Two-dimensional orthogonal (unitary) transformsLet {u[m,n]} be an nn image.

where {ak,l[m,n]}, called an image transform, is a set of complete orthonormal discrete basis functions satisfying the properties:

Orthonormality:

where [a,b] is the 2D delta function, which is one only when a=b=0, and is zero otherwise.

1,0 ],[],[],[

1,0 ],[],[],[

1

0

1

0,

1

0

1

0,

Nnmnmalkvnmu

Nlknmanmulkv

N

k

N

llk

N

m

N

nlk

]','[]','[],[

]','[],[],[

,

1

0

1

0,

','

1

0

1

0,

nnmmnmanma

llkknmanma

lk

N

k

N

llk

lk

N

m

N

nlk

V = {v[k,l]} is called the transformed image.The orthonormal property assures that any expansion of the basis images

will be minimized by the truncated series

When P=Q=N, the error of minimization will be zero.

Separable Unitary TransformsThe number of multiplications and additions required to compute the transform coefficients v[k,l] is O(N4), which is quite excessive.

The dimensionality can be reduced to O(N3) when the transform is restricted to be separable.

NQNPnmalkvnmu lk

P

m

Q

nQP

, ],[],['],[ ,

1

0

1

0,

],[],[' lkvlkv

A transform {ak,l[m,n]} is separable iff for all 0k,l,m,nN1, it can be decomposed as follows:

where A {a[k,m]} and B {b[l,n]} should be unitary matrices themselves, i.e., AAH = AHA = I and BBH = BHB = I .

Often one choose B to be the same as A, so that

Hence, we can simplify the transform as

V = AUAT, and U = A*TVA*

where V = {v[k,l]} and U = {u[m,n]}.

][][ ],[, nbmanma lklk

][],[][],[

][],[][],[

1

0

1

0

1

0

1

0

N

k

N

llk

N

m

N

nlk

nalkvmanmu

nanmumalkv

A more general form: for an MN rectangular image, the transform pair is

V = AMUANT, and U = AM*TUAN*

where AM and AN are MM and NN unitary matrices, respectively. themselves, i.e., AAH = AHA = I and BBH = BHB = I.

These are called two-dimensional separable transforms. The complexity in computing the coefficient image is O(N3).

The computation can be decomposed as computing T=UAT first, and then compute V= AT (for an NN image)

Computing T=UAT requires N2 inner products (of N-point vectors). Each inner product requires N operations, and so in total O(N3).

Similarly, V= AT also requires O(N3) operations, and finally we need O(N3) to compute V.

A closer look at T=UAT:

Let the rows of U be {U1, U2, …, UN}. Then

T=UAT = [U1T, U2

T, …, UNT]TAT = [U1AT, U2AT, …, UNAT] T.

Note that each UiAT (i=1 … N) is a one-dimensional unitary transform. That is, this step performs N one-dimensional transforms for the rows of the image U, obtaining a temporary image T.

Then, the step V= AT performs N 1-D unitary transforms on the columns of T.

Totally, 2N 1-D transforms are performed. Each 1-D transform is of O(N2).

Remember that the two-dimensional DFT is

where

The 2D DFT is separable, and so it can be represented as

V = FUF

where F is the NN matrix with the element of k-th row and n-th element be

F =

1

0

1

0

1

0

1

0

],[1

],[

],[1

],[

N

k

N

l

nlN

mkN

N

m

N

n

lnN

kmN

WWlkvN

nmu

WWnmuN

lkv

)/2( NjN eW

1,0, 1

NnkWN

knN

Fast computation of two-dimensional DFT: According to V = FUF, it can be decomposed as the computation of 2N 1-D DFTs. Each 1-D DFT requires Nlog2N computations.

So, the 2-D DFT can be efficiently implemented in time complexity of O(N2log2N )

2-D DFT is inherent in many properties of 1-D DFT (e.g., conjugate symmetry, shifting, scaling, convolution, etc.). A property not from the 1-D DFT is the rotation property.

Rotation property: if we represent (m,n) and (k,l) in polar coordinate, and

then

That is, the rotation of an image implies the rotation of its DFT.

)sin,cos(),( rrnm )sin,cos(),( wwlk

],[],[ wvruDFT

• Until now, we have assumed that the signals are deterministic, i.e., each value of a sequence is uniquely determined.

• In many situations, the processes that generate signals are so complex as to make precise description of a signal extremely difficult or undesirable.

• A random or stochastic signal is considered to be characterized by a set of probability density functions.

Discrete-time Random Signals(c.f. Oppenheim, et al., 1999)

• Random (or stochastic) process (or signal)– A random process is an indexed family of random varia

bles characterized by a set of probability distribution function.

– A sequence x[n], <n< . Each individual sample x[n] is assumed to be an outcome of some underlying random variable Xn.

– The difference between a single random variable and a random process is that for a random variable the outcome of a random-sampling experiment is mapped into a number, whereas for a random process the outcome is mapped into a sequence.

Stochastic Processes

• Probability density function of x[n]:

• Joint distribution of x[n] and x[m]:

• Eg., x1[n] = Ancos(wn+n), where An and n are random variables for all < n < , then x1[n] is a random process.

Stochastic Processes (continue)

nxp n ,

mxnxp mn ,,,

• x[n] and x[m] are independent iff

• x is a stationary process iff

for all k.• That is, the joint distribution of x[n] and x[m] depen

ds only on the time difference m n.

Independence and Stationary

mxpnxpmxnxp mnmn ,,,,,

mxnxpkmxknxp mnkmkn ,,,,,,

• Particularly, when m = n for a stationary process:

It implies that x[n] is shift invariant.

Stationary (continue)

nxpknxp nkn ,,

• In many of the applications of discrete-time signal processing, random processes serve as models for signals, in the sense that a particular signal can be considered a sample sequence of a random process.

• Although such a signals are unpredictable – making a deterministic approach to signal representation is inappropriate – certain average properties of the ensemble can be determined, given the probability law of the process.

Stochastic Processes vs. Deterministic Signal

• Mean (or average)

denotes the expectation operator

• For independent random variables

Expectation

nnnnx dxnxpxxm

n,

nnnn dxnxpxgxg ,

mnmn yxyx

• Mean squared value

• Variance

Mean Square Value and Variance

nnnn dxnxpxx ,

22}{

2

nxnn mxx var

• Autocorrelation

• Autocovariance

Autocorrelation and Autocovariance

mnmnmn

mnxx

dxdxmxnxpxx

xxn,m

,,,

}{

mn

mn

xxxx

xmxnxx

mmn,m

mxmxn,m

}{

}{

*

• For a stationary process, the autocorrelation is dependent on the time difference m n.

• Thus, for stationary process, we can write

• If we denote the time difference by k, we have

Stationary Process

22

xnx

nxx

mx

xmmn

nknxxxx xxnknk ,

• In many instances, we encounter random processes that are not stationary in the strict sense.

• If the following equations hold, we call the process wide-sense stationary (w. s. s.).

Wide-sense Stationary

22

xnx

nxx

mx

xmmn

nknxxxx xxnknk ,

• For any single sample sequence x[n], define their time average to be

• Similarly, time-average autocorrelation is

Time Averages

L

LnL

nxL

nx12

1lim

nxmnxL

nxmnxL

LnL

12

1lim

• A stationary random process for which time averages equal ensemble averages is called an ergodic process:

Ergodic Process

xmnx

mnxmnx xx

• It is common to assume that a given sequence is a sample sequence of an ergodic random process, so that averages can be computed from a single sequence.

Ergodic Process (continue)

1

0

1

0

22

1

0

1

1

1

L

nL

L

nxx

L

nx

nxmnxL

nxmnx

mnxL

nxL

m

ˆ

ˆ

In practice, we cannot compute with the limits, but instead the quantities. Similar quantities are often computed as estimates of the mean, variance, and autocorrelation.

• Property 1:

Properties of correlation and covariance sequences

ynxmnxy

nmnxy

xnxmnxx

nmnxx

mymxm

yxm

mxmxm

xxm

yxxyxy

xxxxx

mmmm

mmm

2

• Property 2:

• Property 3

Properties of correlation and covariance sequences (continue)

Variance 0

Value SquaredMean 0

2

2

xxy

nxx xE

mmmm

mmmm

xyxyxxxx

xyxyxxxx

• Property 4:

Properties of correlation and covariance sequences (continue)

00

00

2

2

yyxxxy

yyxxxy

m

m

0

0

xxxx

xxxx

m

m

• Property 5:– If

Properties of correlation and covariance sequences (continue)

mm

mm

xxyy

xxyy

0nnn xy

• Since autocorrelation and autocovariance sequences are all (aperiodic) one-dimensional sequences, there Fourier transform exist and are bounded in |w|.

• Let the Fourier transform (DTFT) of the autocorrelation and autocovariance sequences be

Fourier Transform Representation of Random Signals

jw

xyxyjw

xxxx

jwxyxy

jwxxxx

emem

emem

• Consider the inverse Fourier Transforms:

Fourier Transform Representation of Random Signals (continue)

dween

dween

jwnjwxxxx

jwnjwxxxx

2

12

1

• Consequently,

• Denote

to be the power density spectrum (or power spectrum) of the random process x.

Fourier Transform Representation of Random Signals (continue)

dwe

dwenx

jwxxxxx

jwxxxx

2

10

2

10

2

2

jwxxxx ewP

• The total area under power density in [,] is the total energy of the signal.

• Pxx(w) is always real-valued since xx(n) is conjugate symmetric

• For real-valued random processes, Pxx(w) = xx(ejw) is both real and even.

Power Density Spectrum(using Fourier transform to represent a random

signal)

dwePnxnxnx jwxxxx

2

1][][0 2

• Consider a linear system with frequency response h[n]. If x[n] is a stationary random signal with mean mx, then the output y[n] is also a stationary random signal with mean mx equaling to

• Since the input is stationary, mx[nk] = mx , and consequently,

Mean and Linear System

knmkhknxkhnynm xkk

y

xj

kxy meHkhmm 0

DC response

• If x[n] is a real and stationary random signal, the autocorrelation function of the output process is

• Since x[n] is stationary , {x[nk]x[n+mr] } depends only on the time difference m+kr.

Stationary and Linear System

k r

k r

yy

rmnxknxrhkh

rmnxknxrhkh

mnynymnn

,

• Therefore,

The output power density is also stationary.• Generally, for a LTI system having a wide-sense

stationary input, the output is also wide-sense stationary.

Stationary and Linear System (continue)

m

rkmrhkh

mnn

yy

k rxx

yy

,

• By substituting l = rk,

where

• A sequence of the form of chh[l] is called a deterministic autocorrelation sequence.

Power Density Spectrum and Linear System

lhhxx

l kxxyy

lclm

klhkhkhlmm

k

hh klhkhlc

• A sequence of the form of Chh[l], l = rk,

where Chh(ejw) is the Fourier transform of chh[l].

• For real h,

• Thus

Power Density Spectrum and Linear System (continue)

jwxx

jwhh

jwyy eeCe

jwjwjw

hh

hh

eHeHeC

lhlhlc

2jwjwhh eHeC

• We have the relation of the input and the output power spectrums to be the following:

Power Density Spectrum and Linear System (continue)

jwxx

jwjwyy eeHe

2

output theofpower average total2

10

input theofpower average total2

10

22

2

dweeHny

dwenx

jwxx

jwyy

jwxxxx

• Key property: The area over a band of frequencies, wa<|w|<wb, is proportional to the power in the signal in that band.

• To show this, consider an ideal band-pass filter. Let H(ejw) be the frequency of the ideal band pass filter for the band wa<|w|<wb.

• Note that |H(ejw)|2 and xx(ejw) are both even functions. Hence, after ideal low pass filtering,

Power Density Property

dweeHdweeHb

a

a

b

w

w

jwxx

jww

w

jwxx

jw

22

2

1

2

1

outputin power average0 yy

• A white noise signal is a signal for which

– Hence, its samples at different instants of time are uncorrelated.

• The power spectrum of a white noise signal is a constant

• The concept of white noise is very useful in quantization error analysis.

White Noise (or White Gaussian Noise)

2x

jwxx e

mm xxx 2

• The average power of a white-noise is therefore

• White noise is also useful in the representation of random signals whose power spectra are not constant with frequency.– A random signal y[n] with power spectrum yy(ejw) can

be assumed to be the output of a linear time-invariant system with a white-noise input.

White Noise (continue)

22

2

1

2

10 xx

jwxxxx dwdwe

22

xjwjw

yy eHe

• The cross-correlation between input and output of a LTI system:

• That is, the cross-correlation between the input output is the convolution of the impulse response with the input autocorrelation sequence.

Cross-correlation

kxx

k

xy

kmkh

kmnxkhnx

mnynxm

• By further taking the Fourier transform on both sides of the above equation, we have

• This result has a useful application when the input is white noise with variance x

2.

– These equations serve as the bases for estimating the impulse or frequency response of a LTI system if it is possible to observe the output of the system in response to a white-noise input.

Cross-correlation (continue)

jwxx

jwjwxy eeHe

jwx

jwxyxxy eHemhm 22 ,