Manual for K-Notes

Why K-Notes?

Towards the end of preparation, a student has lost the time to revise all the chapters from his /

her class notes / standard text books. This is the reason why K-Notes is specifically intended for

Quick Revision and should not be considered as comprehensive study material.

What are K-Notes?

A 40 page or less notebook for each subject which contains all concepts covered in GATE

Curriculum in a concise manner to aid a student in final stages of his/her preparation. It is highly

useful for both the students as well as working professionals who are preparing for GATE as it

comes handy while traveling long distances.

When do I start using K-Notes?

It is highly recommended to use K-Notes in the last 2 months before GATE Exam

(November end onwards).

How do I use K-Notes?

Once you finish the entire K-Notes for a particular subject, you should practice the respective

Subject Test / Mixed Question Bag containing questions from all the Chapters to make best use

of it.

BASIC CONCEPTS

In continuous time signals independent variable is continuous and thus these signals are

defined for a continuum of values of independent variable.

Discrete time signals are only defined at discrete times and consequently for these

signals the independent variable takes discrete set of values.

Representation of continuous time signals

We use symbol t to denote independent variable for continuous time signal.

These signals can be represented by a wave form as shown below

If possible, these can also be represented by a mathematical function like

x(t) = sin t

Representation of discrete time signal

We use symbol n to denote independent variable for discrete time signal.

These signals can be represented as a series of numbers like

x[n] = [5, 4, 5, 7, 9, 2]

Arrow indicates reference point or x [0]

If possible, we can represent the same by a function like

x[n] = sin n 4 Also these signals can be represented by a wave form as shown below

Energy & Power Signals

Interval ,

Energy of continuous time signal

2 2

T

T

limE x t dt x t dt

T

Energy of discrete time signal

2 2

N

nn N

limE x n x n

T

Power of continuous time signals

2

T

T

lim 1P x t dt

T 2T

Power of discrete time signals

2

N

n N

lim 1P x n

N 2N 1

Signals having non-zero (finite) power and infinite energy are called as Power Signals.

ex. x(t) = sint

Signals having finite (non-zero) energy and zero power are called as Energy Signals.

ex. x[n] = [1, 2, 3, 4]

The bounded signal radiate finite energy and periodic signal radiate finite average

power.

Even & Odd signals

A signal is said to be even if it satisfies the condition

x(t) = x (t) or x [n] = x[n]

A signal is said to be odd if it satisfies the condition

x(t) = x(t) or x [n] = x[n]

Any signal (even those which are neither odd nor even) can be broken into odd & even

parts

Odd Part

0

x t x tx t

2 ;

0

x n x nx n

2

Even Part

ex t x t

x t2

;

e

x n x nx n

2

Periodic and Aperiodic Signals

A signal is said to be periodic with period T or N if

x(t + T) = x(t)

x[n + N] = x[n]

Otherwise, the signals are said to be aperiodic.

Classification of systems

(i) Linear & Non-Linear Systems

For Linearity

if 1 1x t y t

2 2x t y t

then, this condition must be true

1 1 2 2 1 1 2 2x t x t y t y t

Example : y(t) = t x (t) is linear

y[n] = 2x [n] + 3 is non-linear

(ii) Time Invariant & Time-variant Systems

For system to be time-invariant the

following condition must hold true

x(t - ) y(t )

It means that following two realizations must be equivalent

The simplest way to verify this is to check the coefficient of t inside x(t)

eg. y(t) = tx(t) is time invariant

but y(t) = tx(2t) is time variant as coefficient of t in side x(t) is not 1

Otherwise, you need to verify the system equivalence shown above.

(iii) Causal & Non-causal Systems

The output should depend only on present & past values of input.

h t 0 V t 0

For discrete time system

h[n] = 0 V n < 0

(iv) Stable & Unstable Systems

Every Bounded input should produce a bounded output.

K

DT : h k

; CT : h d

(v) LTI systems with or without memory

The output at any time should depend only on value of input at the same time.

For discrete time system

h[n] = 0 V n 0

h[n] = k [n]

For continuous time system

h(t) = 0 V t 0

h [t] = k [t]

(vi) Invertible Systems

The system is invertible if there exists h1(t) such that

Thus h(t) * h1(t) = t

For discrete time, h[n] * h1[n] = n

Shifting and Scaling operations

Shifting

Delay

if

shift the waveform right by the amount of delay

Advance

if

shift the waveform left by the amount of advance

Scaling

Compression

if

Replace upper & lower limit by original limit divided by compression factor

Expansion

if

Replace upper & lower limit by original limit multiplied by expansion factor.

Note : If both scaling and shifting are given in the question .

Ex. x(3t-2)

1. shift the waveform right by the amount of delay

2. Replace upper & lower limit by original limit divided by compression factor

This method is applicable for both continuous and discrete time signal.

LTI system (Linear Time Invariant Systems)

Any continuous time or discrete time system can be represented in terms of impulses.

x t x t d

k

x[n] x k n k

LTI systems are characterized on the basis of Impulse Response h(t) or h[n]

The response of a system with impulse as an input is called as impulse response.

Due to time invariance property of LTI system

if n h n

n k h n k

since

K

x n x k n k

k

y n x k h n k x n * h n

= convolution sum

for continuous time domain

k

y t x h t x t * h t

= convolution integral

The condition for causality of system then becomes

h[n] =0 V n < 0 ; h(t) = 0 V t < 0

Calculating convolution sum

Suppose x [n] = u[n]

h[n] = [1, 2, 5, 7, 9]

Draw plots of both x[n] & h[n]

Flip either x[n] or h[n] about y-axis

Here, we flip x[n]

For calculating y[n], shift x[k] to right by amount n

For y[0]

The only overlapping between the two is at k = 0, 1, 2

y [0] = x[0] h [0] + x [1] h [1] + x [2] h [2]

= 1 x 5 + 1 x 2 + 1 x 1

= 8

For y [1]

y [1] = x [0] h [1] + x [1] h [0] + x [2] h [1] + x [3] h [2]

= 1 x 7 + 1 x 5 + 1 x 1 x 1 x 2 = 15

Similarly, we can calculate all values of y[n]

y[n] = [2, 3, 8, 15, 24, 24..]

Calculating Convolution Integral

Assume x (t) = u (t)

h (t) =

Step 1

Flip either x(t) or h(t)

Here, we flip h(t)

Step 2

Shift h( ) by amount t to the right to calculate y(t) by calculating overlapping between

h t & x

Overlapping area

=

0

1 t

1.1d 1 t

if t < 1

so, overlapping area = 0

if t > 1

overlapping area = 2

y (t) is shown in adjoining figure:

Properties of Convolution Sum

1) Commutative Property

x[n] * h[n] = h[n] * x[n]

2) Distributive Property

y1[n] = x[n] * h1[n]

y2[n] = x[n] * h2[n]

y [n] = y1[n] + y2[n] = x[n] * h1[n] + x2[n]*h2[n]

= x[n] * { h1[n] + h2[n] }

3) Associative Property

{x[n] * h1[n] }* h2[n] = { x[n]* h2[n] } * h1 [n]

Same properties will apply for continuous time domain for convolution integral.

Parallel & Cascade structure of LTI systems

Parallel:

y[n] = x[n] * [h1 [n] + h2 [n]]

Cascade:

y[n] = x[n] * ([h1 [n] + h2 [n]])

Frequency Response

The frequency response of any LTI system is given by its Fourier Transform.

DT: jw jwnn

H e h n e

CT: jwtH jw h t e dt

Group delay & Phase delay

Assuming transfer function of system is H(s)

input is x(t)= jwt

e

Output: j wjwt jwt

H jw e H jw e e

= wt wj

H jw e

w Arg H jw

Group Delay,

g

d ww

dw

Phase Delay, w

ww

Continuous Time Fourier series

Fourier states that any periodic signal can be represented by a set of complex exponential

signals provided that it satisfies Drichlet Conditions.

Drichlet conditions

(i) Over any period x(t) is absolutely integrable

i.e., T

0

x t dt

(ii) In a finite time interval, x(t) has a finite number of maxima & minima

(iii) It should have finite number of discontinuities in the given interval

Note : for distortion less transmission of the of a signal with some finite frequency content

through a continuous time LTI system , the frequency response of the system must satisfy these

two conditions.

1. The magnitude response H( j ) must be constant for all frequencies of interest ;

that is, we must have

H( j ) C

For some constant C

2. For the same frequencies of interest, the phase response arg H( j ) must be linear in frequency, with slope to and intercept zero ; that is, we must have

oarg H( j ) t

Fourier series as generally expressed in 2 forms.

Trigonometric

Exponential

Trigonometric Fourier Series

Analysis equations

T

00

1a x t dt

T

0T

k0

2a x t cos k t dt

T where 0 2 T

0T

k0

2b x t sin k t dt

T

Synthesis equations

0 00 k kk - k k 0 k 0

x t a a cosk t b sink t

Exponential Fourier Series

Analysis equations

0T

jk t

k0

1C x t e dt

T

Synthesis equations

0jk t

Kk

x t C e where 0

2T

Relation between T.F.S. and E.F.S.

0oc a

n nna jb

C2

n nna jb

C2

Important facts about Trigonometric Fourier series

(i) Any odd signal contains only sine terms in Fourier series.

(ii) Any even signal contains only cosine terms in Fourier series.

(iii) For halfwave symmetric signal

Tx t x t2 Only odd harmonics are present

i.e., k = 1, 3, 5.

Properties of complex exponential Fourier Series

(i) Linearity

If kF.S.x t a

kF.S.y t b

then Ax (t) + By (t) F.S. A

ka + B

kb

(ii) Time-shifting

if kF.S.x t a

0 0 0 k-jk tF.S.x t t e a

where

02

T

(iii) Time-Reversal

if kF.S.x t a

kF.S.x t a

For odd signal For even signal

x(t) = x(t) x(t) = x (t)

k k

a a

k k

a a

(iv) Time Scaling

if kF.S.x t a

kF.S.x t a

but 0

is replaced by 0 , though Fourier series coefficients remain same.

(v) Multiplication

if kF.S.x t a

kF.S.y t b

kF.S.z t x t y t c

pk k pP

C b a

= convolution sum

(vi) Parsevals Relation

Energy in time domain = Energy frequency Domain

22

kkT

1x t dt a

T

where kF.S.x t a

Discrete Time Fourier series

For a discrete-time signal, with period N the following equations are used for Fourier

series.

Analysis equations

N

k

2j KnNC x n e

0

2N

0N

k

j KnC x n e

Synthesis equations

0N

K

j Knx n C e

The properties of Fourier series coefficients are same as continuous time Fourier series

with one additional property.

K N KC C

That is, Fourier series coefficients are periodic

IMPORTANT DUALITY

A signal discrete in one domain is periodic in other domain & vice versa.

Example: For continuous Time Fourier Series, x (t) is periodic in time domain & hence Fourier

Series exists where coefficients exist for frequency integral multiple of 0

" " & hence is discrete.

Fourier Transform

Fourier series exists only for periodic signals, Fourier series converges to Fourier Transform

which is continuous as compared to Fourier series which is discrete.

Continuous Time Fourier Transform

Analysis equation

jwtX jw x t e dt

Synthesis equation

jwt1

x t x jw e dw2

Properties of Continuous Time Fourier Transform

Signal Fourier Transform

x(t) X(jw)

y(t) Y(jw)

Ax(t)+By(t) AX(jw)+BY(jw)

x(t-t0)

x*(t) X*(-w)

x(-t) X(-w)

x(at)

x(t)*y(t) X(jw)Y(jw)

jwX(jw)

x(t)y(t)

tx(t)

Ev{x(t)} Re{X(jw)}

Od{x(t)} jIm{X(jw)}

X(t) 2x(-w)

X(w-w0)

Parsevals Relation

2 21

x t x w dw2

0j te X w

jw1X

a a

dx(t)

dt

1X(w) * Y(w)

2

t

x d 1

X jw X 0 wjw

d

j X jwdw

0j te x t

Some common Fourier Transform Pairs

Signal Fourier Transform

0jkw t

kK

a e

0kk

2 a k

0jkw te

02

cos 0w t

0 0

sin 0w t 0 0j

1 2

n

t nT

K

2 2 k

T T

1

1

1, t Tx t

0, t T

12sin T

(sin wt)/t

1, wx

0, w

t 1

u(t)

1

j

0t t 0j te

ate u t ,Re a 0 1a j

Discrete Time Fourier Transform

Analysis equation

j j nn

X e x n e

Synthesis Equation

2

j j n1x n X e e d

2

Properties of Discrete Time Fourier Transform

Signal Fourier Transform

x[n]

y[n]

X periodic with

period 2Y

ax[n] + by [n] aX bY

0x n n

0j ne X

x*[n] X *

0j n

e x n

0X x [n] X

k

x n | k , if n is multiple of k x n

0, is n is not multiple of k

X k

x [n] * y [n]

X Y

n x [n]

dxj

d

Ev x n Re {X( )}

Od {x [n]} j Im {X( )}

Parsevals Relation

2 2

n 2

1x n X d

2

Some common Fourier Transform Pairs

Signal Fourier Transform

k

2jk n

N

K N

a e

kk

2 k2 a

N

0j ne

02 2

cos 0n 0 02 2

0sin n 0 02 2j

x [n] = 1 2 2

1

1

1, n N

x n N0, n N , n

2

and x [n + N] = x [n]

kk

2 k2 a

N

k

n kN

k

2 2 k

N N

11

1, n Nx n

0, n N

1

1sin N2

sin2

sinWn W Wnsinc

n

1, 0 Wx

0, W <

0n n

0j ne

Laplace Transform

Laplace Transform is more general than Fourier Transform but can only be computed in

Region of Convergence (ROC), so it cannot be computed V s

ROC = t

S jw; such that

x t e dt

Laplace transform becomes Fourier transform for 0 , if it lies in ROC.

Analysis Equations

for bilateral Laplace Transform

H(s) = sth t e dt

for unilateral Laplace Transform

H(s) = st

0

h t e dt

Synthesis Equation

x(t) = j

st

j

1x s e ds

2 j

Properties of ROC

(i) ROC consists of a collection of lines parallel to jwaxis in splane.

such that

tx t e dt

(ii) If X (s) is rational, then ROC does not contain any poles.

(iii) If x(t) is of finite duration & absolutely integrable, then ROC is entire s-plane.

(iv) If x(t) is right sided signal (i.e., it is zero before some time) and if Re(s) = 0

is in the

ROC, then all values of s for which Re(s) > 0

are also in ROC.

(v) If x(t) is left sided, (i.e., if it is zero after some time), and if Re (s) = 0

is in ROC, then

all values of s for which Re(s) < 0

are also in ROC.

(vi) If x(t) is twosided signal and if the line Re (S) = 0

is in ROC, then the ROC consists

of a strip in splane include the line Re (S) = 0

(vii) If X(s) is rational, and

x(t) is right sided signal, then ROC is right of right most pole.

x(t) is left sided signal, then ROC is left of left most pole.

Properties of Laplace Transform

Signal Transform ROC

x(t) X(s) R

x1(t)

X1(s)

R1

x2(t) X2(s)

R2

ax1(t) + bx2(t) aX1(s) + bX2(s)

At least R1 R2

0x t t 0st

e X s

R

0s te x t 0X s s Shifted version R [i.e., s is in ROC if 0s s is in R]

x (at)

1 sX

a a

Scaled ROC i.e., s is ROC if

sa is in R

1 2x t * x t 1 2X s X s At least R1 R2

d

x tdt

sX s At least R

tx(t)

dx s

ds

R

t

x d

1

X ss

At least R

Some common Laplace Transform Pairs

Signal Transform ROC

t 1 All s

u(t) 1s

Re {s} > 0

u(t) 1s

Re {s} < 0

n 1tu t

n 1 !

n1

s Re {s} > 0

n 1tu t

n 1 !

n1

s Re {s} < 0

ate u t 1s a

Re {s} > a

- ate u t 1s a

Re {s} < a

n 1att e u t

n 1 !

n

1

s a

Re {s} < a

n 1att e u t

n 1 !

n

1

s a

Re {s} > a

t T sTe All s

0cos t u t

02 2

s

s

Re {s} > 0

0sin t u t

0

0

2 2s

Re {s} > 0

at 0e cos t u t

02 2

s a

s a

Re {s} > a

at 0e sin t u t

0

0

2 2s a

Re {s} > a

Initial and Final Value Theorem

limx 0 sX s initial value

s

limx s X s

s 0 Final value, first stability should be ensured, else final value does

not exist.

Analysis of LTI system using Laplace Transform

Stability

h t dt

; ROC of H(s) should include 0 .

Causality

h(t) = 0, t < 0 i.e., right sided signal

ROC should be right sided

ROC should include Right half plane.

but converse is not true.

Z Transform

It is generalization of Discrete Time Fourier Transform

Analysis Equation

k

k

H z h k z

Synthesis Equation

n 11h[n] H z z dz2 j

Indicates integration around counter clockwise circular contour centered at origin

& with radius r.

ROC for Z-Transform

Z Transform also exists only inside ROC

nn

x n r

is the condition for ROC.

Mapping from s-plane from zplane

The jw-axis is mapped to unit circle in zplane.

Right Half plane is mapped to exterior of unit circle.

Left Half plane is mapped to interior of unit circle.

Properties of ROC

(i) The ROC x(z) consists of a ring in the z plane centered about the origin.

(ii) The ROC does not contain any poles.

(iii) If x[n] is of finite duration, then ROC is the entire z plane except possibility at z = 0

and/or z =

(iv) If x[n] is a right sided sequence and if the circle, | z | = r0 is in the ROC, then all finite

values of z, for which | z | > r0 will also be in ROC.

(v) If x[n] is a left sided sequence, and the circle | z | = r0 is in ROC, then all finite value of

z, for which 0 < | z | < r0 will be in ROC.

(vi) If x[n] is two sided sequence and if circle | z | = r0 is in the ROC. Then ROC will consist

of a ring in z-plane which consist of ring | z | = r0.

(vii) If X (z) is rational and

x[n] is right sided than ROC is outside of outer most pole.

x[n] is left sided then ROC is inside of inner most pole.

(viii) If x[n] is causal, ROC includes z = provided x[n] = 0, n < 0.

If x [n] is anti causal, ROC includes z = 0 provided x [n] = 0, n > 0.

(ix) A causal LTI system with rational system function is stable if all poles inside the unit

circle that is have magnitude, | z | < 1.

Properties of zTransform

Signal Transform ROC

x[n] X(z) xR

1x n 1X z 1R

2x n 2X z 2R

1 2ax n bx n 1 2aX z bX z At least R1 R2

0x n n

0nz X z

Rx with addition or

deletion of origin

0j n

e x n

0jX e z xR

0nz x n

0

zXz

x0z R

x[n] 1X z 1 xz s.t z R

x r , n=rk

w n0, n rk for some r

kX z 1k1k

x xR i.e., z s.t z R

21x n * x n 1 2X z X z At least R1 R2

nx[n] zdX zdz

Rx except addition or

deletion of zero

n

k

x k 1

1X z

1 z

xR z 1

Some common Z Transform pairs

Signal Transform ROC

n 1 All z

u n 1

1

1 z

| z | > 1

u n 1 1

1

1 z

| z | < 1

n m mz All z except 0 (if m > 0) or (if m < 0)

na u n 1

1

1 az

| z | > | a |

na u n 1 1

1

1 az

| z | < | a |

nna u n

1

21

az

1 az

| z | > | a |

nna u n 1

1

21

az

1 az

| z | < | a |

Initial & Final value Theorem

lim

x 0 X zz

Initial value

lim 1x 1 X z

zz 1 Final value

In z transform also, stability must be verified before using final value theorem.

Sampling

Continuous Discrete Time

Time signal signal

Nyquist Sampling Theorem

It states that if sampling frequency is greater than twice the maximum frequency in the

signal for the signal to be recovered from its samples.

MS

w 2w

Note: For this condition signal spectrum should be centered around y-axis.

Band-pass Sampling Theorem

If the signal spectrum is band-pass which means it has minimum & maximum frequency

Lf = lower frequency ; uf = upper frequency

u

u L

fK ,where

f f indicates Greatest Integer function

uS

2fw

K

px (t) = x(t) p(t)

n

p t t nT

T = sampling interval ; px t Sampled signal

x(t) = continuous time signal

pn

x t x t t nT

P1

X w X w *P w2

sk

2P w w kw

T

sPk

1X w X w kw

T

; s2

wT

The spectrum of sampled signal is just repetition of actual spectrum at integral multiples

of sw .

If s Mw 2w , adjacent samples of spectrum overlap, called as aliasing.

Discrete Fourier Transform

DFT of n point sequence is given by:

Analysis equation:

N 1

n 0

j2 kn

NX k x n .e , k = 0, 1, 2., N1

Synthesis equation:

N 1

K 0

j2 kn

N1x n X k eN

, n = 0, 1, 2..., N 1

Each point of DFT require N complex multiplications and (N 1) complex additions.

Therefore, N point DFT will required N2 complex multiplications and N (N 1) complex

additions.

Properties of DFT

Sequence Transform

x[n] X(k)

x1[n] X1[k]

x2[n] X2[k]

x[n + N] = x[n] X(K+N)= X(k)

2 21 1a x n a x n 2 21 1a x k a x k

N 1

N1 2n 0

x n x m n

Where 2 2Nx m n x N m N

21x k x k

Nx n x N n X(N k)

Nx n 2 kj

NX K e

2 nj

NX n e

NX K

x*(n) X*(N k)

1 2x n x n N1 21

X K x kN

Circular convolution of

2 DFT sequences

Parsevals Relation

N 1 N 1

2 2

n 0 K 0

1x n x k

N

Fast Fourier Transform (FFT) Algorithms

These are the algorithms for computing DFT when the size N is a power of 2 or when it is

a power of 4.

Direct computation of DFT is inefficient because it does not exploit the properties of

symmetry and periodicity of the phase factor, 2j

NN

W e

Symmetry property :

NK+

2 KN N

W W

Periodicity property :

K+N KN N

W W

DFT can be expressed as :

N 1

R R In 0

2 kn 2 knX K x n cos x n sin

N N

N 1

I R In 0

2 kn 2 knX K x n sin x n cos

N N

No. of operations required for direct computation of DFT

1) 22N evolutions of trigonometric functions.

2) 24N Real multiplications.

3) 4N (N 1) real additions.

Radix 2 FFT algorithm

There are two types of FFT algorithm:

1) Decimation in time.

2) Decimation in frequency.

Radix 2 algorithm can be implemented over N point DFT sequence if N = n2 .

We divide the given time sequence by 2 till we get the prime factor.

We split the Npoint data sequence into two N

2 point data sequences 1f n and 2f n

corresponding to even numbered and numbered samples of x(n).

NN 1

nk

n 0

X K x n W

, K = 0, 1, ,N 1.

N Nnk nk

n even n odd

x n W x n W

By substituting n = 2r for n even and n = 2r + 1 for n odd,

N N

N N1 12 2

2r 1k2rk

r 0 r 0

X k x 2r W x 2r 1 W

=

N N NN N1 1

2 2rk rk2 2 2

r 0 r 0

x 2r W W x 2r 1 W

But N2

N2

W W .

N

j22 N2 j

N2 2N

2

W e e W

N N1 1

2 2

N N2 2

rk rkN

r 0 r 0

kX k x 2r W W x 2r 1 W

=

N2

1 2

G K W H K

f k f k

, k = 0, 1, , N 1.

Now, computation of X[k] requires 2 2N N N N

22 2 2 2

complex multiplications.

We can further decimate 1f n and 2f n in time. Thus 1f n would result in two N

4

point sequences. 2f n decimation by 2 would also result in two N

4 point sequence.

We can deduce finally that total number of complex multiplication is reduced to

2

Nlog N

2 and the number of complex additions to N

2log N .

FFT computation for 8-point DFT

Order of input is bit reversed, due to decimating twice in time domain.

Number of butterflies in N

2 per stage.

Number of stage is2

log N .

Butterfly computation.

Decimation in frequency

We split the DFT formula into two summations, one of which involves the sum over the first N

2

data points and the second sum involves the last N

2 data points.

Thus,

N1

N 12Kn Kn

N

2

N Nn 0 n

X k x n W x n W

N N1 Nk2 2

Kn Kn2

0

1

N N Nn 0 n

NX k x n .W W x n W

2

Since

2 NkNk jN 2 kj k2

NW e e 1

Therefore,

N1

2Knk

Nn 0

Nx k x n 1 x n W

2

Now, divide X(k) into even and odd numbered samples.

N1

2KnN

2n 0

NX 2k x n x n W

2, k = 0, 1, .

N1

2 .

n KnN

2

N1

2

n 0N

NX 2k 1 x n x n W W

2 , k = 0,1, ,

N1

2 .

Total number of complex multiplications are2

Nlog N

2.

Total Number of complex addition are 2

Nlog N .

Decimation in frequency for N = 8 FFT algorithm

Note :

1) Order of input is normal while order of output is bit reversed.

2) Number of stages is 2

log N .

3) Number of butterflies is N

2 for stage.

Butterfly computation for decimation in frequency.

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