FINAL EXAMINATION

SEMESTER /SESSION

COURSE

:

:

DIGITAL SIGNAL PROCESSING

PROGRAMME

COURSE CODE

:

:

BEJ

BEB 30503

9

TABLE 1: Properties of the N-Sample DFT

Property Signal DFT

Shift x[n – no] Nnkj

DFToekX

2

Shift x[n – 0.5N] kX DFT

k1

Modulation Nknj oenx/2

oDFT kkX

Modulation nxn

1 NkX DFT 5.0

Folding x[– n] kXDFT

Product x[n]y[n] kYkX

NDFTDFT

1

Convolution x[n] y[n] kYkX DFTDFT

Correlation x[n] y[n] kYkX DFTDFT

*

Central Ordinates

1

0

10

N

k

DFT kXN

x ,

1

0

0N

n

DFT nxX

Central Ordinates kX

N

Nx DFT

kN

k

1

0

11

2 (N even),

nxN

X

nN

n

DFT

1

0

12

(N even)

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COURSE CODE

:

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BEB 30503

10

TABLE 2: Properties of z-transform.

Property Signal z-transform

Linearity a1x1(n) + a2x2(n) a1X1(z) + a2X2(z)

Time reversal x(– n) X(z-1)

Time shifting i) x(n – k)

ii) x(n + k)

i) z-kX(z)

ii) zkX(z)

Convolution x1(n)x2(n) X1(z)X2(z)

Correlation

n

xx lnxnxlr 2121 1

2121

zXzXzR xx

Scaling anx(n) X(a-1z)

Differentiation nx(n)

1

1

dz

zdXz or

dz

zdXz

Time

differentiation x(n) – x(n – 1) X(z)(1 - z-1)

Time integration

0k

kX

1z

zzX

Initial value

theorem nx

n 0

lim

zX

z

lim

Final value

theorem nx

n

lim zX

z

z

z

1

1

lim

FINAL EXAMINATION

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11

TABLE 3: Laplace Transform

Signal x(t) Laplace Transform X(s)

δ(t) 1

u(t) s

1

r(t) = tu(t) 2

1

s

t2u(t) 3

2

s

tnu(t) 1

!ns

n

e -αtu(t) s

1

te -αtu(t) 2

1

s

tne -αtu(t) 1

!

n

s

n

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12

TABLE 4: Digital to Digital Frequency Transformations

Form Band

Edges Mapping z Parameters

Lowpass to lowpass ΩC

z

z

1

CD

CD

5.0sin

5.0sin

Lowpass to highpass ΩC z

z

1

CD

CD

5.0cos

5.0cos

Lowpass to bandpass [Ω1, Ω2] 11

2

2

21

2

zAzA

AzAz

125.0tan

5.0tan

DK

12

12

5.0cos

5.0cos

1

21

K

KA

,

1

12

K

KA

Lowpass to bandstop [Ω1, Ω2] 11

2

2

21

2

zAzA

AzAz

K = tan(0.5ΩD)tan[0.5(Ω2 - Ω1)]

12

12

5.0cos

5.0cos

1

21

KA

,

K

KA

1

12

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13

TABLE 5: Direct Analog to Digital Transformations for Bilinear Design.

Form Band Edges Mapping s

Parameters

Lowpass to

lowpass

ΩC

11

zC

z

C = tan(0.5ΩC)

Lowpass to

highpass

ΩC 1

1

z

zC

C = tan(0.5ΩC)

Lowpass to

bandpass

Ω1 < Ω0 < Ω2

112

2

2

zC

zz

C = tan[0.5(Ω2 - Ω1)], = cosΩ0 or

12

12

5.0cos

5.0cos

Lowpass to

bandstop

Ω1 < Ω0 < Ω2 12

12

2

zz

zC

C = tan[0.5(Ω2 - Ω1)], = cosΩ0 or

12

12

5.0cos

5.0cos

TABLE 6: Windows for FIR filter design.

Window Expression wN [n],

– 0.5(N – 1) n 0.5(N – 1)

Boxcar 1

Cosine

1cos

N

n

Riemann sinc ,

1

2

N

nL L > 0

Bartlett

1

21

N

n

Von Hann (Hanning)

1

2cos5.05.0

N

n

Hamming

1

2cos46.054.0

N

n

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Finite Summation Formula

2

1

0

nnn

k

6

121

0

2

nnnn

k

4

122

0

3

nnn

k

1

1 1

0

nn

k

k

, 1

2

1

0 1

11

nnn

k

k nnk

3

22122

0

2

1

12211

nnnn

k

k nnnnk

Infinite Summation Formula

1

1

0k

k

, 1

11k

k

, 1

21 1

k

kk

, 1

32

1

2

1

k

kk

, 1

e

ee

k

k

1

1

, 0