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MODULE 1 > MODULE 1 > Flashcards

MODULE 1 Flashcards

1
Q

CONTINUOUS TIME SIGNAL

A

This is a signal whose independent variable is time “t”.

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1
Q

DISCRETE-TIME SIGNAL

A

This is a signal whose independent variable is discrete-time “n”.

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2
Q

SCALING PROPERTY

A

δ(-at+b)= 1/|a| δ(t-b/a)

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3
Q

INTEGRAL

A

∫_(-∞)^∞▒〖δ(t)dt=1 〗

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4
Q

SAMPLING PROPERTY

A

x(t)δ(t-k)=x(k)δ(t-k)

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5
Q

SIFTING PROPERTY

A

∫_(-∞)^∞▒〖x(t)δ(t-k)dt=x(k)〗

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6
Q

ENERGY IN CT

A

∫_(-∞)^∞▒〖|x(t)|^2 dt 〗

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7
Q

POWER IN CT

A

1/T ∫_T^∞▒〖|x(t)|^2 dt 〗

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8
Q

TRIANGULAR

A

E=(a^2 b)/3

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9
Q

SQUARE

A

〖E= a〗^2 b

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10
Q

CURVE

A

E=(a^2 b)/2

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11
Q

(t-k)u(t-k)

A

r(t-k)

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12
Q

V_RMS

A

√P

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13
Q

∫_0^∞▒〖〖ae〗^(-bt) 〗

A

a/b

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14
Q

f(∝t)

A

E/∝

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15
Q

f(t/∝)

A

|∝|E

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16
Q

f(t±k)

A

E

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17
Q

∝f(t)

A

∝^2 E

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18
Q

g(∝t)

A

P

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19
Q

g(t/∝)

A

P

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20
Q

g(t±k)

A

P

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21
Q

∝g(t)

A

∝^2 P

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22
Q

ENERGY IN DT

A

∑_(-N)^N▒|x(n)|^2

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23
Q

POWER IN DT

A

1/(2N+1) ∑_(-N)^N▒|x(n)|^2

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24
Q

SUMMATION

A

∑_(N=∞)^∞▒〖δ(n)=1〗

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25
Q

TIME SCALING

A

δ(∝n)= δ(n)

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26
Q

SUMMATION PROPERTY

A

∑_(N=n_1)^(n_2)▒〖x(n)δ(n-k)=1〗

NOTE: k ∈ n_1& n_2

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27
Q

MIXED OPERATIONS

A

δ(an+b)= δ(n+b/a)

NOTE: b/a∈Integer

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28
Q

SAMPLING PROPERTY

A

x(n)δ(n-k)=x(k) or 0

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29
Q

EVEN SIGNAL

A

x_e (t)=(x(t)+x(-t))/2

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30
Q

ODD

A

x_o (t)=(x(t)-x(-t))/2

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31
Q

RECTANGULAR SIGNAL

A

x(t)=ARect(t/T)

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32
Q

TRIANGULAR SIGNAL

A

x(t)=ATri(t/T)

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33
Q

SINC SIGNAL

A

sinc(t)=((sin(πt))/(π(t)))

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34
Q

SAMPLING SIGNAL

A

sa(t)=((sin(t))/t)

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35
Q

f_o

A

1/T_o

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36
Q

〖ω〗_o

A

2π/T_o

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37
Q

T_o

A

2π/ω_o

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38
Q

ω_o

A

HCF/LCM

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39
Q

N_o

A

2π/ω_o

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40
Q

STATIC AND DYNAMIC

A

STATIC- present output input
DYNAMIC- past and future

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41
Q

CAUSAL AND NON-CAUSAL

A

CAUSAL- past and present input output
NON-CAUSAL- future

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42
Q

LINEAR AND NON-LINEAR

A

Multiplying coefficients= LINEAR
Time Scaling= LINEAR
Summation of Time Shifted= LINEAR
Added/ subtracted term= NON-LINEAR

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43
Q

TIME VARIANT AND TIME INVARIANT

A

Time Scaling and Time Folding= TV
Coefficient should be constant.
Added/ subtracted time dependent term= TV
Piecewise=TV

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44
Q

STABLE AND UNSTABLE

A

Follow BIBO criteria

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45
Q

CONVOLUTION INTEGRAL

A

y(t)=∫_(-∞)^∞▒x(τ)h(t-τ)dτ

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46
Q

LIMITS OF CONVOLUTION

A

x(t)*h(t)=a+c<t<b+d

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47
Q

TIME SHIFTING IN CV

A

y(t-a-b)=x(t-a)*h(t-b)

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48
Q

AREA OF CONVOLUTION

A

y(t)=∫(-∞)^∞▒〖x(t)h(t)dt=〗 ∫(-∞)^∞▒x(t)dt∙∫_(-∞)^∞▒h(t)dt

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49
Q

TIME SCALING IN CV

A

1/|a| y(t)=x(at)*h(at)

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50
Q

TIME FOLDING IN CV

A

y(-t)=x(-t)*h(-t)

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51
Q

AMPLITUDE SCALING IN CV

A

aby(t)=ax(t)*bh(t)

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52
Q

x(t)*δ(t)

A

x(t)

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53
Q

x(t)*δ(t-k)

A

x(t-k)

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54
Q

x(t)*u(t)

A

∫_(-∝)^∝▒〖x(t)dt〗

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55
Q

u(t)*u(t)

A

r(t)

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56
Q

u(t-a)*u(t-b)

A

r(t-a-b)

57
Q

e^(-at) u(t)*u(t)

A

(1-e^(-at))/a u(t)

58
Q

CONVOLUTION SUMMATION

A

y(n)=∑_(-∞)^∞▒x(k)h(n-k)

59
Q

TIME INVARIANCE IN CS

A

y(n-k)=x(n-k)h(n)=x(n)h(n-k)

60
Q

SUM OF SAMPLES

A

∑(n= -∞)^∞▒y(n) =(∑(n=-∞)^∞▒x(k) )(∑_(n=-∞)^∞▒h(n-k) )

61
Q

LENGTH OF THE OUTPUT

A

L_Y=L_X+L_H-1

62
Q

x(n)*δ(n)

A

x(n)

63
Q

x(n)*δ(n-k)

A

x(n-k)

64
Q

x(n)*u(n)

A

∑_(n=-∝)^∝▒〖x(n)〗

65
Q

u(n)*u(n)

A

r(n+1)

66
Q

u(n-a)*u(n-b)

A

r(n+1-a-b)

67
Q

CONTINUOUS-TIME FOURIER SERIES

A
  • The continuous time Fourier Series (CTFS) refers to a signal processing operation that transforms continuous time signal into discrete frequency signal.
  • The CTFS is used to represent a periodic non-sinusoidal signal into sum of harmonically related sinusoids.
68
Q

TRIGONOMETRIC FOURIER SERIES

A

x(t)= a_0+∑(n=1)^∞▒〖a_n cos(nω_0 t)+∑(n=1)^∞▒〖b_n sin(nω_0 t)〗〗

69
Q

a_0

A

1/t_0 ∫_T▒x(t)dt

70
Q

a_n

A

2/t_0 ∫_T▒x(t)cos(nω_0 t)dt

71
Q

b_n

A

2/t_0 ∫_T▒x(t)sin(nω_0 t)dt

72
Q

ODD SYMMETRY

A

{█(a_0=0@a_n=0@b_n≠0)┤

73
Q

EVEN SMMETRY

A

{█(a_0=0@a_n≠0@b_n=0)┤

74
Q

HALF- WAVE

A

{█(a_0=0@a_n≠b_n≠0 ∀ odd n @〖a_n=b〗_n=0∀ even n)┤

75
Q

POLAR FOURIER SERIES

A

x(t)= d_0+∑_(n=1)^∞▒〖d_n cos(nω_0 t+∅_n ) 〗

76
Q

d_0

A

〖a〗_0=1/t_0 ∫_T▒x(t)dt

77
Q

d_n

A

√(〖a_n〗^2+〖b_n〗^2 )

78
Q

∅_n

A

〖tan〗^(-1) (b_n/a_n )

79
Q

EXPONENTIAL FOURIER SERIES

A

x(t)= c_0+∑_(n=1)^∞▒c_n e^(j〖nω〗_0 t)

80
Q

c_0

A

〖a〗_0=1/t_0 ∫_T▒x(t)dt

81
Q

c_n

A

1/t_0 ∫_T▒〖x(t) e^(-jnω_0 t) dt〗

82
Q

a_n RELATIONSHIP

A

c_n+ c_(-n)

83
Q

b_n RELATIONSHIP

A

j(c_n- c_(-n) )

84
Q

c_n RELATIONSHIP

A

(a_n-jb_n)/2

85
Q

CONTINUOUS-TIME FOURIER TRANSFORM

A
  • The continuous time Fourier transform (CTFT) is a signal processing operation that transforms a continuous-time signal into a continuous frequency signal.
86
Q

MATHEMATICAL FORMULA OF CTFT

A

X(ω)=∫_(-∞)^∞▒〖x(t) e^(-jωt) dt〗

87
Q

e^(-at) u(t)

A

1/(a+jω)

88
Q

e^(-a|t| )

A

2a/(a^2+ω^2 ),a>0

89
Q

〖t ∙e〗^(-at) u(t)

A

1/(a+jω)^2

90
Q

Sgn(t)

A

2/jω

91
Q

Sa(at)

A

π/a rect (ω/2a)

92
Q

rect(t/T)

A

T Sa(ωT/a)

93
Q

TIME SHIFTING IN CTFT

A

x(t-t_0 )↔e^(-jωt_0 ) x(ω)
x(t+t_0 )↔e^(jωt_(0 ) ) x(ω)

94
Q

TIME SCALING IN CTFT

A

x(at)↔1/|a| X(ω/a)

95
Q

TIME FOLDING IN CTFT

A

x(-t)↔ X(-ω)
NOTE: Put negative sign in all ω

96
Q

5FREQUENCY SHIFT IN CTFT

A

e^jωt x(t)↔ X(ω-ω_0 )
e^(-jωt_ ) x(t)↔ X(ω+ω_0 )

97
Q

DUALITY IN CTFT

A

x(t)↔X(ω)
X(t)↔2πx(-ω)

98
Q

TIME DIFFERENTATION IN CTFT

A

d^n/〖dt〗^n [x(t)]=〖(jω)〗^n X(ω)

99
Q

FREQUENCY DIFFERENTATION IN CTFT

A

tx(t)↔j d/dω [X(ω)]

100
Q

MULTIPLICATION PROPERTY IN CTFT

A

〖x_1 (t)x〗_2 (t)↔X_1 (ω)∙X_2 (ω)
〖x_1 (t)∙x〗_2 (t)↔1/2π (X_1 (ω)X(ω))

101
Q

AREA PROPERTY IN CTFT

A

X(ω)=∫(-∞)^∞▒x(t) e^(-jωt) dt
X(0)=∫(-∞)^∞▒x(t) dt=AREA
INVERESE FT
x(t)=1/2π ∫(-∞)^∞▒x(ω) e^jωt dω
x(0)=1/2π ∫(-∞)^∞▒x(ω) dω=AREA

102
Q

PARSEVAL’S THEOREM IN CTFT

A

∫(-∞)^∞▒|x(t)|^2 dt=1/2π ∫(-∞)^∞▒|X(ω)|^2 dω

103
Q

Z TRANSFROM

A
  • Refers to a signal processing operation that transforms a discrete-time signal into a complex frequency signal.
104
Q

MATHEMATICAL FORMULA OF Z

A

X(z)=∑_(n=-∞)^∞▒〖x(n) 〖 z〗^(-n) 〗

105
Q

TIME SHIFTING IN Z TRANS

A

x(n-k)↔z^(-k) X(z)
x(n+k)↔z^k X(z)

106
Q

Z SCALING IN Z TRANS

A

a^n x(n)↔X(z/a)
a^(-n) x(n)↔X(az)

107
Q

TIME FOLDING IN Z TRANS

A

x(-n)↔ x(1/z)

108
Q

CONVOLUTION IN Z TRANS

A

〖x_1 (n)*x〗_2 (n)↔X_1 (z)∙X_2 (z)

109
Q

DIFFERENTIATION IN Z IN Z TRANS

A

nx(n)↔-z d/dz [X(z)]

110
Q

δ(n)

A

1

111
Q

δ(n-k)

A

〖 z〗^(-k)

112
Q

a^n u(n)

A

z/(z-a)

113
Q

a^n u(-n-1)

A

(-1)/(1-a/z)a^n u(-n-1)

114
Q

〖-a〗^n u(-n-1)

A

z/(z-a)

115
Q

REGION OF CONVERGENCE

A
  • This refers to the region in the z-plane where the z-transform converges
116
Q

na^n u(n)

A

az/(z-a)^2

117
Q

Discrete-Time Fourier Transform

A
  • The Discrete-time Fourier Transform (DTFT) refers to a signal processing operation that converts discrete-time signal into continuous frequency signal.
118
Q

MATHEMATICAL FORMULA OD DTFT

A

X(Ω)=∑_(n=-∞)^∞▒〖x(n) 〖 e〗^(-jΩn) 〗

119
Q

TIME SHIFT IN DTFT

A

x(n-k)↔e^(-jΩk) X(Ω)

120
Q

FREQUENCY SHIFT IN DTFT

A

e^jΩn x(n)↔ X(Ω-Ω_0 )
e^(-jΩn_ ) x(n)↔ X(Ω+Ω_0 )

121
Q

HALF PERIOD SHIFT IN DTFT

A

〖(-1)〗^n x(n)↔ X(Ω-π)

122
Q

TIME FOLDING IN DTFT

A

x(-n)↔ X(-Ω)

123
Q

AREA PROPERTY IN DTFT

A

∫(-π)^π▒〖X(Ω) e^(-jΩn( ) ) dΩ〗=2πx(n)
∫_(-π)^π▒X(Ω)dΩ=2πx(0)=AREA

124
Q

SUMMATION PROPERTY IN DTFT

A

∑(n=-∞)^∞▒〖x(n)=〗 X(0)
∑(n=-∞)^∞▒〖〖(-1)〗^n x(n)=〗 X(π)

125
Q

PARSEVALS’S THEOREM IN DTFT

A

∑(-∞)^∞▒|x(n)|^2 =1/2π ∫(-∞)^∞▒|X(Ω)|^2 dΩ

126
Q

FREQUENCY DIFFERENTIATION IN DTFT

A

nx(n)↔j d/dΩ [X(Ω)]

127
Q

CONVOLUTION PROPERTY IN DTFT

A

〖x_1 (n)*x〗_2 (n)↔X_1 (Ω)∙X_2 (Ω)

128
Q

MULTIPLICATION PROPERTY IN DTFT

A

〖x_1 (n)∙x〗_2 (n)↔1/2π (X_1 (Ω)⊛X(Ω))

129
Q

Discrete Fourier Transform

A
  • The Discrete Fourier Transform (DFT) refers to a signal processing operation that converts n-sample, periodic discrete-time signal into discrete frequency signal.
  • Strictly for finite length sequence.
  • The input should be periodic.
130
Q

MATHEMATICAL FORMULA OF DFT

A

X(k)=∑_(n=0)^(n-1)▒〖x(n) 〖 e〗^(-jΩ_k n) 〗
WHERE,
Ω_k=2π/N k

131
Q

TWIDDLE FACTOR

A

〖 e〗^(-jΩ_k )=W_n

132
Q

W_4

A

〖 e〗^(-j 2π/4)=-j

133
Q

W_8

A

〖 e〗^(-j 2π/8)=(√2-j√2)/2

134
Q

CIRCULAR SHIFT

A

〖x(n-n_0 )〗_(mod N)↔〖 e〗^(-jΩ_k n_0 ) X(k)

135
Q

FREQUENCY SHIFT IN DFT

A

〖 e〗^(-jΩ_k0 n) x(n)↔X(k-k_0 )

136
Q

CIRCUILAR FOLDING

A

〖x(-n)〗(mod N)↔〖X(-k)〗(mod N)

137
Q

SYMMETRY

A

X(k)=X*(N-K)

138
Q

CIRCULAR CONVOLUTION

A

x_1 (n)⊛x(n)↔X_1 (k)X(k)

139
Q

MULTIPLICATION PROPERTY IN DFT

A

〖x_1 (n)∙x〗_2 (n)↔1/N (X_1 (k)⊛X(k))

140
Q

FREQUENCY INTERPOLATION

A

Interpolation in f.d ↔ replication in f.d

141
Q

EXISTENCE OF DTFT

A
  • If ROC includes the unit circle
  • Unit circle r=1

MODULE 1 (12 decks)

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