Formula sheet: Basic System Properties | Digital Signal Processing - Electronics and Communication Engineering (ECE) PDF Download

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Digital Signal Pro cessing: Basic Syst em Prop erties
F orm ula Sheet for GA TE
Basic System Prop erties
• Linearit y : A system is linear if it satisfies sup erp osition (additivit y and homogeneit y).
– A dditivit y : F or inputs x
1
[n] and x
2
[n] , and outputs y
1
[n] = T{x
1
[n]} , y
2
[n] =
T{x
2
[n]} ,
T{x
1
[n]+x
2
[n]} =y
1
[n]+y
2
[n]
– Homogeneit y : F or a scalar a ,
T{ax[n]} =aT{x[n]}
– Sup erp osition: Com bines b oth,
T{ax
1
[n]+bx
2
[n]} =ay
1
[n]+by
2
[n]
• Time-In v ariance : A system is time-in v arian t if a time shift in the input causes an
iden tical shift in the output.
If y[n] =T{x[n]}, then y[n-n
0
] =T{x[n-n
0
]}
• Causalit y : A system is causal if the output at an y time n dep ends only on the input
at time n and earlier.
y[n] dep ends only on x[k] for k=n
F or a system describ ed b y impulse resp onse h[n] , it is causal if:
h[n] = 0 for n< 0
• Stabilit y : A system is stable (Bounded-Input Bounded-Output, BIBO) if ev ery
b ounded input pro duces a b ounded output.
|x[n]|=M
x
<8 =? |y[n]|=M
y
<8
F or a linear time-in v arian t (L TI) system, stabilit y re quires:
8
?
n=-8
|h[n]|<8
where h[n] is the impulse resp onse.
• Memory : A system has memory if the output at time n dep ends on input v alues
other than x[n] .
Memoryless: y[n] =f(x[n])
With memory: y[n] dep ends on x[k] for k?=n
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Digital Signal Pro cessing: Basic Syst em Prop erties
F orm ula Sheet for GA TE
Basic System Prop erties
• Linearit y : A system is linear if it satisfies sup erp osition (additivit y and homogeneit y).
– A dditivit y : F or inputs x
1
[n] and x
2
[n] , and outputs y
1
[n] = T{x
1
[n]} , y
2
[n] =
T{x
2
[n]} ,
T{x
1
[n]+x
2
[n]} =y
1
[n]+y
2
[n]
– Homogeneit y : F or a scalar a ,
T{ax[n]} =aT{x[n]}
– Sup erp osition: Com bines b oth,
T{ax
1
[n]+bx
2
[n]} =ay
1
[n]+by
2
[n]
• Time-In v ariance : A system is time-in v arian t if a time shift in the input causes an
iden tical shift in the output.
If y[n] =T{x[n]}, then y[n-n
0
] =T{x[n-n
0
]}
• Causalit y : A system is causal if the output at an y time n dep ends only on the input
at time n and earlier.
y[n] dep ends only on x[k] for k=n
F or a system describ ed b y impulse resp onse h[n] , it is causal if:
h[n] = 0 for n< 0
• Stabilit y : A system is stable (Bounded-Input Bounded-Output, BIBO) if ev ery
b ounded input pro duces a b ounded output.
|x[n]|=M
x
<8 =? |y[n]|=M
y
<8
F or a linear time-in v arian t (L TI) system, stabilit y re quires:
8
?
n=-8
|h[n]|<8
where h[n] is the impulse resp onse.
• Memory : A system has memory if the output at time n dep ends on input v alues
other than x[n] .
Memoryless: y[n] =f(x[n])
With memory: y[n] dep ends on x[k] for k?=n
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• In v ertibilit y : A system is in v ertible if there exists an in v erse system that reco v ers
the original input from the output.
y[n] =T{x[n]}, x[n] =T
-1
{y[n]}
F or L TI systems, in v ertibilit y requires a unique impulse resp onse h
-1
[n] suc h that:
h[n]*h
-1
[n] =d[n]
where d[n] is the unit impulse.
L TI System Represen tation
• Impulse Resp onse : Output of an L TI system to a unit impulse d[n] .
y[n] =x[n]*h[n] =
8
?
k=-8
x[k]h[n-k]
• Difference Equation : General form for an L TI system.
N
?
k=0
a
k
y[n-k] =
M
?
m=0
b
m
x[n-m]
where a
k
and b
m
are co e?icien ts, N and M are the orders of the output and input
terms.
Key Notes
• L TI Systems : Most GA TE questions fo cus on linear time-in v arian t systems.
• T esting Prop erties : Use test signals (e.g., x[n] =d[n] , x[n] =u[n] ) to c hec k prop er-
ties.
• Con v olution : Key for L TI systems; ensure familiarit y with con v olution sum.
• Units : All signals and indices (n , k ) are dimensionless in discrete-time systems.
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