Formula Sheets: Introduction to Signals & Systems | Signals and Systems - Electrical Engineering (EE) PDF Download

 Page 1


Formula Sheet for Introduction to Signals and
Systems – GATE
1. Signal Classi?cations
• Continuous-Time Signal: x(t), de?ned for all t.
• Discrete-Time Signal: x[n], de?ned at discrete times n.
• Periodic Signal:
x(t+T) = x(t), x[n+N] = x[n]
where T: Period (continuous), N: Period (discrete).
• Fundamental Frequency:
f
0
=
1
T
, ?
0
=
2p
T
(continuous), ? 0
=
2p
N
(discrete)
• Even and Odd Signals:
x(-t) = x(t) (even), x(-t) =-x(t) (odd)
x
e
(t) =
x(t)+x(-t)
2
, x
o
(t) =
x(t)-x(-t)
2
2. Basic Signals
• Unit Impulse (Dirac Delta):
d(t) :
?
8
-8
d(t)dt = 1, d[n] : d[n] = 1 if n = 0, else 0
• Unit Step:
u(t) : u(t) = 1 for t= 0, else 0, u[n] : u[n] = 1 for n= 0, else 0
• Unit Ramp:
r(t) = tu(t), r[n] = nu[n]
• Exponential Signal:
x(t) = Ae
st
, x[n] = Az
n
, s = s +j?, z = re
j? 3. System Classi?cations
• Linearity: System is linear if it satis?es superposition:
T{ax
1
(t)+bx
2
(t)} = aT{x
1
(t)}+bT{x
2
(t)}
• Time-Invariance: System is time-invariant if:
y(t-t
0
) = T{x(t-t
0
)}
1
Page 2


Formula Sheet for Introduction to Signals and
Systems – GATE
1. Signal Classi?cations
• Continuous-Time Signal: x(t), de?ned for all t.
• Discrete-Time Signal: x[n], de?ned at discrete times n.
• Periodic Signal:
x(t+T) = x(t), x[n+N] = x[n]
where T: Period (continuous), N: Period (discrete).
• Fundamental Frequency:
f
0
=
1
T
, ?
0
=
2p
T
(continuous), ? 0
=
2p
N
(discrete)
• Even and Odd Signals:
x(-t) = x(t) (even), x(-t) =-x(t) (odd)
x
e
(t) =
x(t)+x(-t)
2
, x
o
(t) =
x(t)-x(-t)
2
2. Basic Signals
• Unit Impulse (Dirac Delta):
d(t) :
?
8
-8
d(t)dt = 1, d[n] : d[n] = 1 if n = 0, else 0
• Unit Step:
u(t) : u(t) = 1 for t= 0, else 0, u[n] : u[n] = 1 for n= 0, else 0
• Unit Ramp:
r(t) = tu(t), r[n] = nu[n]
• Exponential Signal:
x(t) = Ae
st
, x[n] = Az
n
, s = s +j?, z = re
j? 3. System Classi?cations
• Linearity: System is linear if it satis?es superposition:
T{ax
1
(t)+bx
2
(t)} = aT{x
1
(t)}+bT{x
2
(t)}
• Time-Invariance: System is time-invariant if:
y(t-t
0
) = T{x(t-t
0
)}
1
• Causality: Output depends only on present and past inputs:
y(t) = f(x(t),t = t)
• Stability (BIBO): Bounded input produces bounded output:
|x(t)|= M
x
<8 =? |y(t)|= M
y
<8
4. Signal Operations
• Time Shifting:
x(t-t
0
), x[n-n
0
]
• Time Scaling:
x(at), x[an] (discrete, a integer)
• Time Reversal:
x(-t), x[-n]
• Amplitude Scaling:
Ax(t), Ax[n]
5. Convolution
• Continuous-Time Convolution:
y(t) = x(t)*h(t) =
?
8
-8
x(t)h(t-t)dt
• Discrete-Time Convolution:
y[n] = x[n]*h[n] =
8
?
k=-8
x[k]h[n-k]
• Properties:
– Commutative: x*h = h*x
– Associative: (x*h
1
)*h
2
= x*(h
1
*h
2
)
– Distributive: x*(h
1
+h
2
) = x*h
1
+x*h
2
6. Energy and Power
• Signal Energy:
E =
?
8
-8
|x(t)|
2
dt, E =
8
?
n=-8
|x[n]|
2
• Signal Power (Periodic/Finite):
P =
1
T
?
T
|x(t)|
2
dt, P =
1
N
N-1
?
n=0
|x[n]|
2
2
Page 3


Formula Sheet for Introduction to Signals and
Systems – GATE
1. Signal Classi?cations
• Continuous-Time Signal: x(t), de?ned for all t.
• Discrete-Time Signal: x[n], de?ned at discrete times n.
• Periodic Signal:
x(t+T) = x(t), x[n+N] = x[n]
where T: Period (continuous), N: Period (discrete).
• Fundamental Frequency:
f
0
=
1
T
, ?
0
=
2p
T
(continuous), ? 0
=
2p
N
(discrete)
• Even and Odd Signals:
x(-t) = x(t) (even), x(-t) =-x(t) (odd)
x
e
(t) =
x(t)+x(-t)
2
, x
o
(t) =
x(t)-x(-t)
2
2. Basic Signals
• Unit Impulse (Dirac Delta):
d(t) :
?
8
-8
d(t)dt = 1, d[n] : d[n] = 1 if n = 0, else 0
• Unit Step:
u(t) : u(t) = 1 for t= 0, else 0, u[n] : u[n] = 1 for n= 0, else 0
• Unit Ramp:
r(t) = tu(t), r[n] = nu[n]
• Exponential Signal:
x(t) = Ae
st
, x[n] = Az
n
, s = s +j?, z = re
j? 3. System Classi?cations
• Linearity: System is linear if it satis?es superposition:
T{ax
1
(t)+bx
2
(t)} = aT{x
1
(t)}+bT{x
2
(t)}
• Time-Invariance: System is time-invariant if:
y(t-t
0
) = T{x(t-t
0
)}
1
• Causality: Output depends only on present and past inputs:
y(t) = f(x(t),t = t)
• Stability (BIBO): Bounded input produces bounded output:
|x(t)|= M
x
<8 =? |y(t)|= M
y
<8
4. Signal Operations
• Time Shifting:
x(t-t
0
), x[n-n
0
]
• Time Scaling:
x(at), x[an] (discrete, a integer)
• Time Reversal:
x(-t), x[-n]
• Amplitude Scaling:
Ax(t), Ax[n]
5. Convolution
• Continuous-Time Convolution:
y(t) = x(t)*h(t) =
?
8
-8
x(t)h(t-t)dt
• Discrete-Time Convolution:
y[n] = x[n]*h[n] =
8
?
k=-8
x[k]h[n-k]
• Properties:
– Commutative: x*h = h*x
– Associative: (x*h
1
)*h
2
= x*(h
1
*h
2
)
– Distributive: x*(h
1
+h
2
) = x*h
1
+x*h
2
6. Energy and Power
• Signal Energy:
E =
?
8
-8
|x(t)|
2
dt, E =
8
?
n=-8
|x[n]|
2
• Signal Power (Periodic/Finite):
P =
1
T
?
T
|x(t)|
2
dt, P =
1
N
N-1
?
n=0
|x[n]|
2
2
• Non-Periodic Power:
P = lim
T?8
1
2T
?
T
-T
|x(t)|
2
dt, P = lim
N?8
1
2N +1
N
?
n=-N
|x[n]|
2
7. System Response
• Impulse Response: h(t) or h[n], output for input d(t) or d[n].
• Output of LTI System:
y(t) = x(t)*h(t), y[n] = x[n]*h[n]
• Step Response: Output for input u(t) or u[n]:
s(t) =
?
t
-8
h(t)dt, s[n] =
n
?
k=-8
h[k]
8. Symmetry Properties
• Even-Odd Decomposition:
x(t) = x
e
(t)+x
o
(t), x[n] = x
e
[n]+x
o
[n]
• Convolution with Even/Odd Signals:
x
e
*h
e
is even, x
o
*h
o
is even, x
e
*h
o
is odd
9. Correlation
• Cross-Correlation:
R
xy
(t) =
?
8
-8
x(t)y(t+t)dt, R
xy
[n] =
8
?
k=-8
x[k]y[n+k]
• Auto-Correlation:
R
xx
(t) = x(t)*x(-t), R
xx
[n] = x[n]*x[-n]
10. Design Considerations
• Signal Types: Use unit step, impulse, or exponential for system testing.
• System Properties: Check linearity, time-invariance, causality, and stability.
• Applications: Signal processing, control systems, communication systems.
• Convolution Simpli?cation: Use symmetry or periodicity to reduce computa-
tion.
3