MODULE 1 Flashcards
CONTINUOUS TIME SIGNAL
This is a signal whose independent variable is time “t”.
DISCRETE-TIME SIGNAL
This is a signal whose independent variable is discrete-time “n”.
SCALING PROPERTY
δ(-at+b)= 1/|a| δ(t-b/a)
INTEGRAL
∫_(-∞)^∞▒〖δ(t)dt=1 〗
SAMPLING PROPERTY
x(t)δ(t-k)=x(k)δ(t-k)
SIFTING PROPERTY
∫_(-∞)^∞▒〖x(t)δ(t-k)dt=x(k)〗
ENERGY IN CT
∫_(-∞)^∞▒〖|x(t)|^2 dt 〗
POWER IN CT
1/T ∫_T^∞▒〖|x(t)|^2 dt 〗
TRIANGULAR
E=(a^2 b)/3
SQUARE
〖E= a〗^2 b
CURVE
E=(a^2 b)/2
(t-k)u(t-k)
r(t-k)
V_RMS
√P
∫_0^∞▒〖〖ae〗^(-bt) 〗
a/b
f(∝t)
E/∝
f(t/∝)
|∝|E
f(t±k)
E
∝f(t)
∝^2 E
g(∝t)
P
g(t/∝)
P
g(t±k)
P
∝g(t)
∝^2 P
ENERGY IN DT
∑_(-N)^N▒|x(n)|^2
POWER IN DT
1/(2N+1) ∑_(-N)^N▒|x(n)|^2
SUMMATION
∑_(N=∞)^∞▒〖δ(n)=1〗
TIME SCALING
δ(∝n)= δ(n)
SUMMATION PROPERTY
∑_(N=n_1)^(n_2)▒〖x(n)δ(n-k)=1〗
NOTE: k ∈ n_1& n_2
MIXED OPERATIONS
δ(an+b)= δ(n+b/a)
NOTE: b/a∈Integer
SAMPLING PROPERTY
x(n)δ(n-k)=x(k) or 0
EVEN SIGNAL
x_e (t)=(x(t)+x(-t))/2
ODD
x_o (t)=(x(t)-x(-t))/2
RECTANGULAR SIGNAL
x(t)=ARect(t/T)
TRIANGULAR SIGNAL
x(t)=ATri(t/T)
SINC SIGNAL
sinc(t)=((sin(πt))/(π(t)))
SAMPLING SIGNAL
sa(t)=((sin(t))/t)
f_o
1/T_o
〖ω〗_o
2π/T_o
T_o
2π/ω_o
ω_o
HCF/LCM
N_o
2π/ω_o
STATIC AND DYNAMIC
STATIC- present output input
DYNAMIC- past and future
CAUSAL AND NON-CAUSAL
CAUSAL- past and present input output
NON-CAUSAL- future
LINEAR AND NON-LINEAR
Multiplying coefficients= LINEAR
Time Scaling= LINEAR
Summation of Time Shifted= LINEAR
Added/ subtracted term= NON-LINEAR
TIME VARIANT AND TIME INVARIANT
Time Scaling and Time Folding= TV
Coefficient should be constant.
Added/ subtracted time dependent term= TV
Piecewise=TV
STABLE AND UNSTABLE
Follow BIBO criteria
CONVOLUTION INTEGRAL
y(t)=∫_(-∞)^∞▒x(τ)h(t-τ)dτ
LIMITS OF CONVOLUTION
x(t)*h(t)=a+c<t<b+d
TIME SHIFTING IN CV
y(t-a-b)=x(t-a)*h(t-b)
AREA OF CONVOLUTION
y(t)=∫(-∞)^∞▒〖x(t)h(t)dt=〗 ∫(-∞)^∞▒x(t)dt∙∫_(-∞)^∞▒h(t)dt
TIME SCALING IN CV
1/|a| y(t)=x(at)*h(at)
TIME FOLDING IN CV
y(-t)=x(-t)*h(-t)
AMPLITUDE SCALING IN CV
aby(t)=ax(t)*bh(t)
x(t)*δ(t)
x(t)
x(t)*δ(t-k)
x(t-k)
x(t)*u(t)
∫_(-∝)^∝▒〖x(t)dt〗
u(t)*u(t)
r(t)
u(t-a)*u(t-b)
r(t-a-b)
e^(-at) u(t)*u(t)
(1-e^(-at))/a u(t)
CONVOLUTION SUMMATION
y(n)=∑_(-∞)^∞▒x(k)h(n-k)
TIME INVARIANCE IN CS
y(n-k)=x(n-k)h(n)=x(n)h(n-k)
SUM OF SAMPLES
∑(n= -∞)^∞▒y(n) =(∑(n=-∞)^∞▒x(k) )(∑_(n=-∞)^∞▒h(n-k) )
LENGTH OF THE OUTPUT
L_Y=L_X+L_H-1
x(n)*δ(n)
x(n)
x(n)*δ(n-k)
x(n-k)
x(n)*u(n)
∑_(n=-∝)^∝▒〖x(n)〗
u(n)*u(n)
r(n+1)
u(n-a)*u(n-b)
r(n+1-a-b)
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.
TRIGONOMETRIC FOURIER SERIES
x(t)= a_0+∑(n=1)^∞▒〖a_n cos(nω_0 t)+∑(n=1)^∞▒〖b_n sin(nω_0 t)〗〗
a_0
1/t_0 ∫_T▒x(t)dt
a_n
2/t_0 ∫_T▒x(t)cos(nω_0 t)dt
b_n
2/t_0 ∫_T▒x(t)sin(nω_0 t)dt
ODD SYMMETRY
{█(a_0=0@a_n=0@b_n≠0)┤
EVEN SMMETRY
{█(a_0=0@a_n≠0@b_n=0)┤
HALF- WAVE
{█(a_0=0@a_n≠b_n≠0 ∀ odd n @〖a_n=b〗_n=0∀ even n)┤
POLAR FOURIER SERIES
x(t)= d_0+∑_(n=1)^∞▒〖d_n cos(nω_0 t+∅_n ) 〗
d_0
〖a〗_0=1/t_0 ∫_T▒x(t)dt
d_n
√(〖a_n〗^2+〖b_n〗^2 )
∅_n
〖tan〗^(-1) (b_n/a_n )
EXPONENTIAL FOURIER SERIES
x(t)= c_0+∑_(n=1)^∞▒c_n e^(j〖nω〗_0 t)
c_0
〖a〗_0=1/t_0 ∫_T▒x(t)dt
c_n
1/t_0 ∫_T▒〖x(t) e^(-jnω_0 t) dt〗
a_n RELATIONSHIP
c_n+ c_(-n)
b_n RELATIONSHIP
j(c_n- c_(-n) )
c_n RELATIONSHIP
(a_n-jb_n)/2
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.
MATHEMATICAL FORMULA OF CTFT
X(ω)=∫_(-∞)^∞▒〖x(t) e^(-jωt) dt〗
e^(-at) u(t)
1/(a+jω)
e^(-a|t| )
2a/(a^2+ω^2 ),a>0
〖t ∙e〗^(-at) u(t)
1/(a+jω)^2
Sgn(t)
2/jω
Sa(at)
π/a rect (ω/2a)
rect(t/T)
T Sa(ωT/a)
TIME SHIFTING IN CTFT
x(t-t_0 )↔e^(-jωt_0 ) x(ω)
x(t+t_0 )↔e^(jωt_(0 ) ) x(ω)
TIME SCALING IN CTFT
x(at)↔1/|a| X(ω/a)
TIME FOLDING IN CTFT
x(-t)↔ X(-ω)
NOTE: Put negative sign in all ω
5FREQUENCY SHIFT IN CTFT
e^jωt x(t)↔ X(ω-ω_0 )
e^(-jωt_ ) x(t)↔ X(ω+ω_0 )
DUALITY IN CTFT
x(t)↔X(ω)
X(t)↔2πx(-ω)
TIME DIFFERENTATION IN CTFT
d^n/〖dt〗^n [x(t)]=〖(jω)〗^n X(ω)
FREQUENCY DIFFERENTATION IN CTFT
tx(t)↔j d/dω [X(ω)]
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))
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
PARSEVAL’S THEOREM IN CTFT
∫(-∞)^∞▒|x(t)|^2 dt=1/2π ∫(-∞)^∞▒|X(ω)|^2 dω
Z TRANSFROM
- Refers to a signal processing operation that transforms a discrete-time signal into a complex frequency signal.
MATHEMATICAL FORMULA OF Z
X(z)=∑_(n=-∞)^∞▒〖x(n) 〖 z〗^(-n) 〗
TIME SHIFTING IN Z TRANS
x(n-k)↔z^(-k) X(z)
x(n+k)↔z^k X(z)
Z SCALING IN Z TRANS
a^n x(n)↔X(z/a)
a^(-n) x(n)↔X(az)
TIME FOLDING IN Z TRANS
x(-n)↔ x(1/z)
CONVOLUTION IN Z TRANS
〖x_1 (n)*x〗_2 (n)↔X_1 (z)∙X_2 (z)
DIFFERENTIATION IN Z IN Z TRANS
nx(n)↔-z d/dz [X(z)]
δ(n)
1
δ(n-k)
〖 z〗^(-k)
a^n u(n)
z/(z-a)
a^n u(-n-1)
(-1)/(1-a/z)a^n u(-n-1)
〖-a〗^n u(-n-1)
z/(z-a)
REGION OF CONVERGENCE
- This refers to the region in the z-plane where the z-transform converges
na^n u(n)
az/(z-a)^2
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.
MATHEMATICAL FORMULA OD DTFT
X(Ω)=∑_(n=-∞)^∞▒〖x(n) 〖 e〗^(-jΩn) 〗
TIME SHIFT IN DTFT
x(n-k)↔e^(-jΩk) X(Ω)
FREQUENCY SHIFT IN DTFT
e^jΩn x(n)↔ X(Ω-Ω_0 )
e^(-jΩn_ ) x(n)↔ X(Ω+Ω_0 )
HALF PERIOD SHIFT IN DTFT
〖(-1)〗^n x(n)↔ X(Ω-π)
TIME FOLDING IN DTFT
x(-n)↔ X(-Ω)
AREA PROPERTY IN DTFT
∫(-π)^π▒〖X(Ω) e^(-jΩn( ) ) dΩ〗=2πx(n)
∫_(-π)^π▒X(Ω)dΩ=2πx(0)=AREA
SUMMATION PROPERTY IN DTFT
∑(n=-∞)^∞▒〖x(n)=〗 X(0)
∑(n=-∞)^∞▒〖〖(-1)〗^n x(n)=〗 X(π)
PARSEVALS’S THEOREM IN DTFT
∑(-∞)^∞▒|x(n)|^2 =1/2π ∫(-∞)^∞▒|X(Ω)|^2 dΩ
FREQUENCY DIFFERENTIATION IN DTFT
nx(n)↔j d/dΩ [X(Ω)]
CONVOLUTION PROPERTY IN DTFT
〖x_1 (n)*x〗_2 (n)↔X_1 (Ω)∙X_2 (Ω)
MULTIPLICATION PROPERTY IN DTFT
〖x_1 (n)∙x〗_2 (n)↔1/2π (X_1 (Ω)⊛X(Ω))
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.
MATHEMATICAL FORMULA OF DFT
X(k)=∑_(n=0)^(n-1)▒〖x(n) 〖 e〗^(-jΩ_k n) 〗
WHERE,
Ω_k=2π/N k
TWIDDLE FACTOR
〖 e〗^(-jΩ_k )=W_n
W_4
〖 e〗^(-j 2π/4)=-j
W_8
〖 e〗^(-j 2π/8)=(√2-j√2)/2
CIRCULAR SHIFT
〖x(n-n_0 )〗_(mod N)↔〖 e〗^(-jΩ_k n_0 ) X(k)
FREQUENCY SHIFT IN DFT
〖 e〗^(-jΩ_k0 n) x(n)↔X(k-k_0 )
CIRCUILAR FOLDING
〖x(-n)〗(mod N)↔〖X(-k)〗(mod N)
SYMMETRY
X(k)=X*(N-K)
CIRCULAR CONVOLUTION
x_1 (n)⊛x(n)↔X_1 (k)X(k)
MULTIPLICATION PROPERTY IN DFT
〖x_1 (n)∙x〗_2 (n)↔1/N (X_1 (k)⊛X(k))
FREQUENCY INTERPOLATION
Interpolation in f.d ↔ replication in f.d
EXISTENCE OF DTFT
- If ROC includes the unit circle
- Unit circle r=1
