Formula sheet: Operations on Signals | Digital Signal Processing - Electronics and Communication Engineering (ECE) PDF Download

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Digital Signal Pro cessing: Op erations on Signals
F orm ula Sheet for GA TE
Basic Signal Op erations
• Time Shifting : Shifts the signal in time.
y(n) =x(n-n
0
) (Discrete-time, dela y b y n
0
)
y(t) =x(t-t
0
) (Con tin uous-time, dela y b y t
0
)
• Time Scaling : Compresses or expands the signal.
y(n) =x(an) (Discrete-time, a> 1 compresses, 0<a< 1 expands)
y(t) =x(at) (Con tin uous-time, a> 1 compresses, 0<a< 1 expands)
• Time Rev ersal : Rev erses the signal.
y(n) =x(-n) (Discrete-time)
y(t) =x(-t) (Con tin uous-time)
• Amplitude Scaling : Scales the signal amplitude.
y(n) =Ax(n), y(t) =Ax(t) (A is a constan t)
Arithmetic Op erations
• A ddition : Sum of t w o signals.
y(n) =x
1
(n)+x
2
(n) (Discrete-time)
y(t) =x
1
(t)+x
2
(t) (Con tin uous-time)
• Multiplication : Pro duct of t w o signals.
y(n) =x
1
(n)·x
2
(n) (Discrete-time)
y(t) =x
1
(t)·x
2
(t) (Con tin uous-time)
Con v olution
• Discrete-Time Con v olution : Output of a linear time-in v arian t (L TI) system.
y(n) =x(n)*h(n) =
8
?
k=-8
x(k)h(n-k)
where x(n) is the input signal, h(n) is the impulse resp onse.
1
Page 2


Digital Signal Pro cessing: Op erations on Signals
F orm ula Sheet for GA TE
Basic Signal Op erations
• Time Shifting : Shifts the signal in time.
y(n) =x(n-n
0
) (Discrete-time, dela y b y n
0
)
y(t) =x(t-t
0
) (Con tin uous-time, dela y b y t
0
)
• Time Scaling : Compresses or expands the signal.
y(n) =x(an) (Discrete-time, a> 1 compresses, 0<a< 1 expands)
y(t) =x(at) (Con tin uous-time, a> 1 compresses, 0<a< 1 expands)
• Time Rev ersal : Rev erses the signal.
y(n) =x(-n) (Discrete-time)
y(t) =x(-t) (Con tin uous-time)
• Amplitude Scaling : Scales the signal amplitude.
y(n) =Ax(n), y(t) =Ax(t) (A is a constan t)
Arithmetic Op erations
• A ddition : Sum of t w o signals.
y(n) =x
1
(n)+x
2
(n) (Discrete-time)
y(t) =x
1
(t)+x
2
(t) (Con tin uous-time)
• Multiplication : Pro duct of t w o signals.
y(n) =x
1
(n)·x
2
(n) (Discrete-time)
y(t) =x
1
(t)·x
2
(t) (Con tin uous-time)
Con v olution
• Discrete-Time Con v olution : Output of a linear time-in v arian t (L TI) system.
y(n) =x(n)*h(n) =
8
?
k=-8
x(k)h(n-k)
where x(n) is the input signal, h(n) is the impulse resp onse.
1
• Con tin uous-Time Con v olution :
y(t) =x(t)*h(t) =
?
8
-8
x(t)h(t-t)dt
• Prop erties of Con v olution :
Comm utativ e : x(n)*h(n) =h(n)*x(n)
Asso ciativ e : [x(n)*h
1
(n)]*h
2
(n) =x(n)*[h
1
(n)*h
2
(n)]
Distributiv e : x(n)*[h
1
(n)+h
2
(n)] =x(n)*h
1
(n)+x(n)*h
2
(n)
Correlation
• Discrete-Time Cross-Correlation : Measures similarit y b et w een t w o signals.
R
xy
(l) =
8
?
n=-8
x(n)y(n+l)
where l is the lag.
• Con tin uous-Time Cross-Correlation :
R
xy
(t) =
?
8
-8
x(t)y(t+t)dt
• A uto-Correlation : Correlation of a signal with itself.
R
xx
(l) =
8
?
n=-8
x(n)x(n+l) (Discrete-time)
R
xx
(t) =
?
8
-8
x(t)x(t+t)dt (Con tin uous-time)
Signal Energy and P o w er
• Energy (Discrete-Time) : F or energy signals.
E =
8
?
n=-8
|x(n)|
2
• Energy (Con tin uous-Time) :
E =
?
8
-8
|x(t)|
2
dt
• P o w er (Discrete-Time) : F or p o w er signals.
P = lim
N?8
1
2N +1
N
?
n=-N
|x(n)|
2
• P o w er (Con tin uous-Time) :
P = lim
T?8
1
2T
?
T
-T
|x(t)|
2
dt
2
Page 3


Digital Signal Pro cessing: Op erations on Signals
F orm ula Sheet for GA TE
Basic Signal Op erations
• Time Shifting : Shifts the signal in time.
y(n) =x(n-n
0
) (Discrete-time, dela y b y n
0
)
y(t) =x(t-t
0
) (Con tin uous-time, dela y b y t
0
)
• Time Scaling : Compresses or expands the signal.
y(n) =x(an) (Discrete-time, a> 1 compresses, 0<a< 1 expands)
y(t) =x(at) (Con tin uous-time, a> 1 compresses, 0<a< 1 expands)
• Time Rev ersal : Rev erses the signal.
y(n) =x(-n) (Discrete-time)
y(t) =x(-t) (Con tin uous-time)
• Amplitude Scaling : Scales the signal amplitude.
y(n) =Ax(n), y(t) =Ax(t) (A is a constan t)
Arithmetic Op erations
• A ddition : Sum of t w o signals.
y(n) =x
1
(n)+x
2
(n) (Discrete-time)
y(t) =x
1
(t)+x
2
(t) (Con tin uous-time)
• Multiplication : Pro duct of t w o signals.
y(n) =x
1
(n)·x
2
(n) (Discrete-time)
y(t) =x
1
(t)·x
2
(t) (Con tin uous-time)
Con v olution
• Discrete-Time Con v olution : Output of a linear time-in v arian t (L TI) system.
y(n) =x(n)*h(n) =
8
?
k=-8
x(k)h(n-k)
where x(n) is the input signal, h(n) is the impulse resp onse.
1
• Con tin uous-Time Con v olution :
y(t) =x(t)*h(t) =
?
8
-8
x(t)h(t-t)dt
• Prop erties of Con v olution :
Comm utativ e : x(n)*h(n) =h(n)*x(n)
Asso ciativ e : [x(n)*h
1
(n)]*h
2
(n) =x(n)*[h
1
(n)*h
2
(n)]
Distributiv e : x(n)*[h
1
(n)+h
2
(n)] =x(n)*h
1
(n)+x(n)*h
2
(n)
Correlation
• Discrete-Time Cross-Correlation : Measures similarit y b et w een t w o signals.
R
xy
(l) =
8
?
n=-8
x(n)y(n+l)
where l is the lag.
• Con tin uous-Time Cross-Correlation :
R
xy
(t) =
?
8
-8
x(t)y(t+t)dt
• A uto-Correlation : Correlation of a signal with itself.
R
xx
(l) =
8
?
n=-8
x(n)x(n+l) (Discrete-time)
R
xx
(t) =
?
8
-8
x(t)x(t+t)dt (Con tin uous-time)
Signal Energy and P o w er
• Energy (Discrete-Time) : F or energy signals.
E =
8
?
n=-8
|x(n)|
2
• Energy (Con tin uous-Time) :
E =
?
8
-8
|x(t)|
2
dt
• P o w er (Discrete-Time) : F or p o w er signals.
P = lim
N?8
1
2N +1
N
?
n=-N
|x(n)|
2
• P o w er (Con tin uous-Time) :
P = lim
T?8
1
2T
?
T
-T
|x(t)|
2
dt
2
Key Notes
• Use consisten t notation: n for discrete-time, t for con tin uous-time.
• Con v olution is cen tral to L TI systems; practice with unit impulse and step functions.
• Correlation is k ey for signal similarit y and system iden tification.
• Ensure prop er limits for summations/in tegrals in GA TE problems.
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