MODULE 1 Flashcards
(142 cards)
1
Q
CONTINUOUS TIME SIGNAL
A
This is a signal whose independent variable is time “t”.
1
Q
DISCRETE-TIME SIGNAL
A
This is a signal whose independent variable is discrete-time “n”.
2
Q
SCALING PROPERTY
A
δ(-at+b)= 1/|a| δ(t-b/a)
3
Q
INTEGRAL
A
∫_(-∞)^∞▒〖δ(t)dt=1 〗
4
Q
SAMPLING PROPERTY
A
x(t)δ(t-k)=x(k)δ(t-k)
5
Q
SIFTING PROPERTY
A
∫_(-∞)^∞▒〖x(t)δ(t-k)dt=x(k)〗
6
Q
ENERGY IN CT
A
∫_(-∞)^∞▒〖|x(t)|^2 dt 〗
7
Q
POWER IN CT
A
1/T ∫_T^∞▒〖|x(t)|^2 dt 〗
8
Q
TRIANGULAR
A
E=(a^2 b)/3
9
Q
SQUARE
A
〖E= a〗^2 b
10
Q
CURVE
A
E=(a^2 b)/2
11
Q
(t-k)u(t-k)
A
r(t-k)
12
Q
V_RMS
A
√P
13
Q
∫_0^∞▒〖〖ae〗^(-bt) 〗
A
a/b
14
Q
f(∝t)
A
E/∝
15
Q
f(t/∝)
A
|∝|E
16
Q
f(t±k)
A
E
17
Q
∝f(t)
A
∝^2 E
18
Q
g(∝t)
A
P
19
Q
g(t/∝)
A
P
20
Q
g(t±k)
A
P
21
Q
∝g(t)
A
∝^2 P
22
Q
ENERGY IN DT
A
∑_(-N)^N▒|x(n)|^2
23
Q
POWER IN DT
A
1/(2N+1) ∑_(-N)^N▒|x(n)|^2
24
SUMMATION
∑_(N=∞)^∞▒〖δ(n)=1〗
25
TIME SCALING
δ(∝n)= δ(n)
26
SUMMATION PROPERTY
∑_(N=n_1)^(n_2)▒〖x(n)δ(n-k)=1〗
NOTE: k ∈ n_1& n_2
27
MIXED OPERATIONS
δ(an+b)= δ(n+b/a)
NOTE: b/a∈Integer
28
SAMPLING PROPERTY
x(n)δ(n-k)=x(k) or 0
29
EVEN SIGNAL
x_e (t)=(x(t)+x(-t))/2
30
ODD
x_o (t)=(x(t)-x(-t))/2
31
RECTANGULAR SIGNAL
x(t)=ARect(t/T)
32
TRIANGULAR SIGNAL
x(t)=ATri(t/T)
33
SINC SIGNAL
sinc(t)=((sin(πt))/(π(t)))
34
SAMPLING SIGNAL
sa(t)=((sin(t))/t)
35
f_o
1/T_o
36
〖ω〗_o
2π/T_o
37
T_o
2π/ω_o
38
ω_o
HCF/LCM
39
N_o
2π/ω_o
40
STATIC AND DYNAMIC
STATIC- present output input
DYNAMIC- past and future
41
CAUSAL AND NON-CAUSAL
CAUSAL- past and present input output
NON-CAUSAL- future
42
LINEAR AND NON-LINEAR
Multiplying coefficients= LINEAR
Time Scaling= LINEAR
Summation of Time Shifted= LINEAR
Added/ subtracted term= NON-LINEAR
43
TIME VARIANT AND TIME INVARIANT
Time Scaling and Time Folding= TV
Coefficient should be constant.
Added/ subtracted time dependent term= TV
Piecewise=TV
44
STABLE AND UNSTABLE
Follow BIBO criteria
45
CONVOLUTION INTEGRAL
y(t)=∫_(-∞)^∞▒x(τ)h(t-τ)dτ
46
LIMITS OF CONVOLUTION
x(t)*h(t)=a+c
47
TIME SHIFTING IN CV
y(t-a-b)=x(t-a)*h(t-b)
48
AREA OF CONVOLUTION
y(t)=∫_(-∞)^∞▒〖x(t)h(t)dt=〗 ∫_(-∞)^∞▒x(t)dt∙∫_(-∞)^∞▒h(t)dt
49
TIME SCALING IN CV
1/|a| y(t)=x(at)*h(at)
50
TIME FOLDING IN CV
y(-t)=x(-t)*h(-t)
51
AMPLITUDE SCALING IN CV
aby(t)=ax(t)*bh(t)
52
x(t)*δ(t)
x(t)
53
x(t)*δ(t-k)
x(t-k)
54
x(t)*u(t)
∫_(-∝)^∝▒〖x(t)dt〗
55
u(t)*u(t)
r(t)
56
u(t-a)*u(t-b)
r(t-a-b)
57
e^(-at) u(t)*u(t)
(1-e^(-at))/a u(t)
58
CONVOLUTION SUMMATION
y(n)=∑_(-∞)^∞▒x(k)h(n-k)
59
TIME INVARIANCE IN CS
y(n-k)=x(n-k)*h(n)=x(n)*h(n-k)
60
SUM OF SAMPLES
∑_(n= -∞)^∞▒y(n) =(∑_(n=-∞)^∞▒x(k) )(∑_(n=-∞)^∞▒h(n-k) )
61
LENGTH OF THE OUTPUT
L_Y=L_X+L_H-1
62
x(n)*δ(n)
x(n)
63
x(n)*δ(n-k)
x(n-k)
64
x(n)*u(n)
∑_(n=-∝)^∝▒〖x(n)〗
65
u(n)*u(n)
r(n+1)
66
u(n-a)*u(n-b)
r(n+1-a-b)
67
CONTINUOUS-TIME FOURIER SERIES
- 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
TRIGONOMETRIC FOURIER SERIES
x(t)= a_0+∑_(n=1)^∞▒〖a_n cos(nω_0 t)+∑_(n=1)^∞▒〖b_n sin(nω_0 t)〗〗
69
a_0
1/t_0 ∫_T▒x(t)dt
70
a_n
2/t_0 ∫_T▒x(t)cos(nω_0 t)dt
71
b_n
2/t_0 ∫_T▒x(t)sin(nω_0 t)dt
72
ODD SYMMETRY
{█(a_0=0@a_n=0@b_n≠0)┤
73
EVEN SMMETRY
{█(a_0=0@a_n≠0@b_n=0)┤
74
HALF- WAVE
{█(a_0=0@a_n≠b_n≠0 ∀ odd n @〖a_n=b〗_n=0∀ even n)┤
75
POLAR FOURIER SERIES
x(t)= d_0+∑_(n=1)^∞▒〖d_n cos(nω_0 t+∅_n ) 〗
76
d_0
〖a〗_0=1/t_0 ∫_T▒x(t)dt
77
d_n
√(〖a_n〗^2+〖b_n〗^2 )
78
∅_n
〖tan〗^(-1) (b_n/a_n )
79
EXPONENTIAL FOURIER SERIES
x(t)= c_0+∑_(n=1)^∞▒c_n e^(j〖nω〗_0 t)
80
c_0
〖a〗_0=1/t_0 ∫_T▒x(t)dt
81
c_n
1/t_0 ∫_T▒〖x(t) e^(-jnω_0 t) dt〗
82
a_n RELATIONSHIP
c_n+ c_(-n)
83
b_n RELATIONSHIP
j(c_n- c_(-n) )
84
c_n RELATIONSHIP
(a_n-jb_n)/2
85
CONTINUOUS-TIME FOURIER TRANSFORM
- The continuous time Fourier transform (CTFT) is a signal processing operation that transforms a continuous-time signal into a continuous frequency signal.
86
MATHEMATICAL FORMULA OF CTFT
X(ω)=∫_(-∞)^∞▒〖x(t) e^(-jωt) dt〗
87
e^(-at) u(t)
1/(a+jω)
88
e^(-a|t| )
2a/(a^2+ω^2 ),a>0
89
〖t ∙e〗^(-at) u(t)
1/(a+jω)^2
90
Sgn(t)
2/jω
91
Sa(at)
π/a rect (ω/2a)
92
rect(t/T)
T Sa(ωT/a)
93
TIME SHIFTING IN CTFT
x(t-t_0 )↔e^(-jωt_0 ) x(ω)
x(t+t_0 )↔e^(jωt_(0 ) ) x(ω)
94
TIME SCALING IN CTFT
x(at)↔1/|a| X(ω/a)
95
TIME FOLDING IN CTFT
x(-t)↔ X(-ω)
NOTE: Put negative sign in all ω
96
5FREQUENCY SHIFT IN CTFT
e^jωt x(t)↔ X(ω-ω_0 )
e^(-jωt_ ) x(t)↔ X(ω+ω_0 )
97
DUALITY IN CTFT
x(t)↔X(ω)
X(t)↔2πx(-ω)
98
TIME DIFFERENTATION IN CTFT
d^n/〖dt〗^n [x(t)]=〖(jω)〗^n X(ω)
99
FREQUENCY DIFFERENTATION IN CTFT
tx(t)↔j d/dω [X(ω)]
100
MULTIPLICATION PROPERTY IN CTFT
〖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
AREA PROPERTY IN CTFT
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
PARSEVAL’S THEOREM IN CTFT
∫_(-∞)^∞▒|x(t)|^2 dt=1/2π ∫_(-∞)^∞▒|X(ω)|^2 dω
103
Z TRANSFROM
- Refers to a signal processing operation that transforms a discrete-time signal into a complex frequency signal.
104
MATHEMATICAL FORMULA OF Z
X(z)=∑_(n=-∞)^∞▒〖x(n) 〖 z〗^(-n) 〗
105
TIME SHIFTING IN Z TRANS
x(n-k)↔z^(-k) X(z)
x(n+k)↔z^k X(z)
106
Z SCALING IN Z TRANS
a^n x(n)↔X(z/a)
a^(-n) x(n)↔X(az)
107
TIME FOLDING IN Z TRANS
x(-n)↔ x(1/z)
108
CONVOLUTION IN Z TRANS
〖x_1 (n)*x〗_2 (n)↔X_1 (z)∙X_2 (z)
109
DIFFERENTIATION IN Z IN Z TRANS
nx(n)↔-z d/dz [X(z)]
110
δ(n)
1
111
δ(n-k)
〖 z〗^(-k)
112
a^n u(n)
z/(z-a)
113
a^n u(-n-1)
(-1)/(1-a/z)a^n u(-n-1)
114
〖-a〗^n u(-n-1)
z/(z-a)
115
REGION OF CONVERGENCE
- This refers to the region in the z-plane where the z-transform converges
116
na^n u(n)
az/(z-a)^2
117
Discrete-Time Fourier Transform
- The Discrete-time Fourier Transform (DTFT) refers to a signal processing operation that converts discrete-time signal into continuous frequency signal.
118
MATHEMATICAL FORMULA OD DTFT
X(Ω)=∑_(n=-∞)^∞▒〖x(n) 〖 e〗^(-jΩn) 〗
119
TIME SHIFT IN DTFT
x(n-k)↔e^(-jΩk) X(Ω)
120
FREQUENCY SHIFT IN DTFT
e^jΩn x(n)↔ X(Ω-Ω_0 )
e^(-jΩn_ ) x(n)↔ X(Ω+Ω_0 )
121
HALF PERIOD SHIFT IN DTFT
〖(-1)〗^n x(n)↔ X(Ω-π)
122
TIME FOLDING IN DTFT
x(-n)↔ X(-Ω)
123
AREA PROPERTY IN DTFT
∫_(-π)^π▒〖X(Ω) e^(-jΩn_( ) ) dΩ〗=2πx(n)
∫_(-π)^π▒X(Ω)dΩ=2πx(0)=AREA
124
SUMMATION PROPERTY IN DTFT
∑_(n=-∞)^∞▒〖x(n)=〗 X(0)
∑_(n=-∞)^∞▒〖〖(-1)〗^n x(n)=〗 X(π)
125
PARSEVALS’S THEOREM IN DTFT
∑_(-∞)^∞▒|x(n)|^2 =1/2π ∫_(-∞)^∞▒|X(Ω)|^2 dΩ
126
FREQUENCY DIFFERENTIATION IN DTFT
nx(n)↔j d/dΩ [X(Ω)]
127
CONVOLUTION PROPERTY IN DTFT
〖x_1 (n)*x〗_2 (n)↔X_1 (Ω)∙X_2 (Ω)
128
MULTIPLICATION PROPERTY IN DTFT
〖x_1 (n)∙x〗_2 (n)↔1/2π (X_1 (Ω)⊛X(Ω))
129
Discrete Fourier Transform
- 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
MATHEMATICAL FORMULA OF DFT
X(k)=∑_(n=0)^(n-1)▒〖x(n) 〖 e〗^(-jΩ_k n) 〗
WHERE,
Ω_k=2π/N k
131
TWIDDLE FACTOR
〖 e〗^(-jΩ_k )=W_n
132
W_4
〖 e〗^(-j 2π/4)=-j
133
W_8
〖 e〗^(-j 2π/8)=(√2-j√2)/2
134
CIRCULAR SHIFT
〖x(n-n_0 )〗_(mod N)↔〖 e〗^(-jΩ_k n_0 ) X(k)
135
FREQUENCY SHIFT IN DFT
〖 e〗^(-jΩ_k0 n) x(n)↔X(k-k_0 )
136
CIRCUILAR FOLDING
〖x(-n)〗_(mod N)↔〖X(-k)〗_(mod N)
137
SYMMETRY
X(k)=X*(N-K)
138
CIRCULAR CONVOLUTION
x_1 (n)⊛x(n)↔X_1 (k)X(k)
139
MULTIPLICATION PROPERTY IN DFT
〖x_1 (n)∙x〗_2 (n)↔1/N (X_1 (k)⊛X(k))
140
FREQUENCY INTERPOLATION
Interpolation in f.d ↔ replication in f.d
141
EXISTENCE OF DTFT
- If ROC includes the unit circle
- Unit circle r=1
