Formula sheet: Introduction to Digital Signal Processing | Digital Signal Processing - Electronics and Communication Engineering (ECE) PDF Download

 Page 1


In tro duction to Digital Signal Pro cessing F orm ula
Sheet for GA TE
Discrete-Time Signals
• Discrete-Time Signal Represen tation :
x[n] =x(nT
s
)
where x[n] is the discrete-time signal, T
s
is the sampling p erio d (s ), n is the sample
index.
• Sampling Theorem (Nyquist-Shannon) :
f
s
= 2f
max
where f
s
=
1
Ts
is the sampling frequency (Hz ), f
max
is the maxim um frequenc y of the
analog signal.
• Aliasing : Occurs if f
s
< 2f
max
, causing frequency comp onen ts ab o v e
fs
2
to fold bac k.
Z-T ransform
• Z-T ransform Definition :
X(z) =
8
X
n=-8
x[n]z
-n
where z is a complex v ariable, region of con v ergence (R OC) determines stabilit y .
• In v erse Z-T ransform : T ypically computed using partial fraction expansion or con-
tour in tegration.
• Prop erties :
– Linearit y : Z{ax[n]+by[n]} =aX(z)+bY(z)
– Time Shift : Z{x[n-k]} =z
-k
X(z)
– Con v olution : Z{x[n]*h[n]} =X(z)H(z)
• P ole-Zero Plot : Stabilit y if R OC includes the unit circle (|z| = 1 ).
Discrete F ourier T ransform (DFT)
• DFT Definition :
X[k] =
N-1
X
n=0
x[n]e
-j
2p
N
kn
, k = 0,1,...,N -1
where N is the n um b er of samples, X[k] is the k -th frequency bin.
1
Page 2


In tro duction to Digital Signal Pro cessing F orm ula
Sheet for GA TE
Discrete-Time Signals
• Discrete-Time Signal Represen tation :
x[n] =x(nT
s
)
where x[n] is the discrete-time signal, T
s
is the sampling p erio d (s ), n is the sample
index.
• Sampling Theorem (Nyquist-Shannon) :
f
s
= 2f
max
where f
s
=
1
Ts
is the sampling frequency (Hz ), f
max
is the maxim um frequenc y of the
analog signal.
• Aliasing : Occurs if f
s
< 2f
max
, causing frequency comp onen ts ab o v e
fs
2
to fold bac k.
Z-T ransform
• Z-T ransform Definition :
X(z) =
8
X
n=-8
x[n]z
-n
where z is a complex v ariable, region of con v ergence (R OC) determines stabilit y .
• In v erse Z-T ransform : T ypically computed using partial fraction expansion or con-
tour in tegration.
• Prop erties :
– Linearit y : Z{ax[n]+by[n]} =aX(z)+bY(z)
– Time Shift : Z{x[n-k]} =z
-k
X(z)
– Con v olution : Z{x[n]*h[n]} =X(z)H(z)
• P ole-Zero Plot : Stabilit y if R OC includes the unit circle (|z| = 1 ).
Discrete F ourier T ransform (DFT)
• DFT Definition :
X[k] =
N-1
X
n=0
x[n]e
-j
2p
N
kn
, k = 0,1,...,N -1
where N is the n um b er of samples, X[k] is the k -th frequency bin.
1
• In v erse DFT :
x[n] =
1
N
N-1
X
k=0
X[k]e
j
2p
N
kn
, n = 0,1,...,N -1
• F requency Resolution :
?f =
f
s
N
(Hz )
• Prop erties :
– Linearit y : ax[n]+by[n]?aX[k]+bY[k]
– Circular Con v olution : x[n]*y[n]?X[k]Y[k]
– P arsev al’s Theorem :
P
N-1
n=0
|x[n]|
2
=
1
N
P
N-1
k=0
|X[k]|
2
Digital Filters
• Difference Equation (FIR/I IR Filters) :
y[n] =
M-1
X
k=0
b
k
x[n-k]-
N-1
X
k=1
a
k
y[n-k]
where b
k
are feedforw ard co e ?icien ts, a
k
are feedbac k co e?icien ts.
• T ransfer F unction :
H(z) =
P
M-1
k=0
b
k
z
-k
1+
P
N-1
k=1
a
k
z
-k
• F requency Resp onse :
H(e
j?
) =H(z)


z=e
j?
, ? = 2p
f
f
s
where ? is the normalized frequency (radians).
• Stabilit y : I IR filter is stable if all p oles of H(z) lie inside the unit circle.
Signal Pro cessing in Systems
• Con v olution :
y[n] =x[n]*h[n] =
8
X
k=-8
x[k]h[n-k]
where h[n] is the impulse resp onse.
• Signal Energy :
E =
8
X
n=-8
|x[n]|
2
• Signal P o w er :
P = lim
N?8
1
2N +1
N
X
n=-N
|x[n]|
2
2
Page 3


In tro duction to Digital Signal Pro cessing F orm ula
Sheet for GA TE
Discrete-Time Signals
• Discrete-Time Signal Represen tation :
x[n] =x(nT
s
)
where x[n] is the discrete-time signal, T
s
is the sampling p erio d (s ), n is the sample
index.
• Sampling Theorem (Nyquist-Shannon) :
f
s
= 2f
max
where f
s
=
1
Ts
is the sampling frequency (Hz ), f
max
is the maxim um frequenc y of the
analog signal.
• Aliasing : Occurs if f
s
< 2f
max
, causing frequency comp onen ts ab o v e
fs
2
to fold bac k.
Z-T ransform
• Z-T ransform Definition :
X(z) =
8
X
n=-8
x[n]z
-n
where z is a complex v ariable, region of con v ergence (R OC) determines stabilit y .
• In v erse Z-T ransform : T ypically computed using partial fraction expansion or con-
tour in tegration.
• Prop erties :
– Linearit y : Z{ax[n]+by[n]} =aX(z)+bY(z)
– Time Shift : Z{x[n-k]} =z
-k
X(z)
– Con v olution : Z{x[n]*h[n]} =X(z)H(z)
• P ole-Zero Plot : Stabilit y if R OC includes the unit circle (|z| = 1 ).
Discrete F ourier T ransform (DFT)
• DFT Definition :
X[k] =
N-1
X
n=0
x[n]e
-j
2p
N
kn
, k = 0,1,...,N -1
where N is the n um b er of samples, X[k] is the k -th frequency bin.
1
• In v erse DFT :
x[n] =
1
N
N-1
X
k=0
X[k]e
j
2p
N
kn
, n = 0,1,...,N -1
• F requency Resolution :
?f =
f
s
N
(Hz )
• Prop erties :
– Linearit y : ax[n]+by[n]?aX[k]+bY[k]
– Circular Con v olution : x[n]*y[n]?X[k]Y[k]
– P arsev al’s Theorem :
P
N-1
n=0
|x[n]|
2
=
1
N
P
N-1
k=0
|X[k]|
2
Digital Filters
• Difference Equation (FIR/I IR Filters) :
y[n] =
M-1
X
k=0
b
k
x[n-k]-
N-1
X
k=1
a
k
y[n-k]
where b
k
are feedforw ard co e ?icien ts, a
k
are feedbac k co e?icien ts.
• T ransfer F unction :
H(z) =
P
M-1
k=0
b
k
z
-k
1+
P
N-1
k=1
a
k
z
-k
• F requency Resp onse :
H(e
j?
) =H(z)


z=e
j?
, ? = 2p
f
f
s
where ? is the normalized frequency (radians).
• Stabilit y : I IR filter is stable if all p oles of H(z) lie inside the unit circle.
Signal Pro cessing in Systems
• Con v olution :
y[n] =x[n]*h[n] =
8
X
k=-8
x[k]h[n-k]
where h[n] is the impulse resp onse.
• Signal Energy :
E =
8
X
n=-8
|x[n]|
2
• Signal P o w er :
P = lim
N?8
1
2N +1
N
X
n=-N
|x[n]|
2
2
Key Notes
• Ensure f
s
= 2f
max
to a v oid aliasing i n sampling.
• Z-transform is used for system analysis; DFT for frequency analysis.
• FIR filters are inheren tly stable; I IR filters require p ole analysis.
• Use SI units: frequency in Hz , time in s , energy in J .
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