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Discrete Systems 37
In DSP we will say that the system processes an input signal into an output
signal. Discrete systems are broadly classified into linear and nonlinear
systems. We will deal mostly with linear systems.
2.2.1 LINEAR SYSTEMS
A discrete system T [·] isalinear operator L[·] ifand only if L[·] satisfies
the principle of superposition, namely,
L[a 1 x 1 (n)+a 2 x 2 (n)] = a 1 L[x 1 (n)] + a 2 L[x 2 (n)], ∀a 1 ,a 2 ,x 1 (n),x 2 (n)
(2.10)
Using (2.3) and (2.10), the output y(n) ofalinear system to an arbitrary
input x(n) isgiven by
[ ∞
]
∑
∞∑
y(n) =L[x(n)] = L x(k) δ(n − k) = x(k)L[δ(n − k)]
n=−∞
n=−∞
The response L[δ(n − k)] can be interpreted as the response of a linear
system at time n due to a unit sample (a well-known sequence) at time k.
It is called an impulse response and is denoted by h(n, k). The output
then is given by the superposition summation
∞∑
y(n) = x(k)h(n, k) (2.11)
n=−∞
The computation of (2.11) requires the time-varying impulse response
h(n, k), which in practice is not very convenient. Therefore time-invariant
systems are widely used in DSP.
□ EXAMPLE 2.5 Determine whether the following systems are linear:
1. y(n) =T [x(n)] =3x 2 (n)
2. y(n) =2x(n − 2)+5
3. y(n) =x(n +1)− x(n − 1)
Solution
Let y 1 (n) = T [ x 1 (n) ] and y 2 (n) = T [ x 2 (n) ] .Wewill determine the
response of each system to the linear combination a 1 x 1 (n)+a 2 x 2 (n) and
check whether it is equal to the linear combination a 1 x 1 (n) +a 2 x 2 (n)
where a 1 and a 2 are arbitrary constants.
1. y(n) =T [x(n)] = 3x 2 (n): Consider
T [ a 1 x 1 (n)+a 2 x 2 (n) ] =3[a 1 x 1 (n)+a 2 x 2 (n)] 2
=3a 2 1x 2 1(n)+3a 2 2x 2 2(n)+6a 1 a 2 x 1 (n)x 2 (n)
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