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3. LINEAR TIME INVARIANT SYSTEM (LTI SYSTEM)

3.1. Analog System

f(t) is the input and g(t) is the output.

f(t) -> g(t)
f(t+) -> g(t+)      time invariant

f1(t) -> g1(t) => af1(t) + bf2(t) -> ag1(t) + bg2(t)     linear
f2 (t) -> g2 (t)

Let h be the impulse response of the system,

We can write

h(t) = h(s) (t-s) ds.

We describe the system by L and derive the formula for the output y(t) when the input is x(t).

L is linear and time invariant

x(t) = x(s) (t-s) ds
y(t) = L [x(t)]
= L [x(s) (t-s) ds]
= x(s) L [(t-s)] ds       (linearity)
= x(s) h(t-s) ds      (time invariance)
Thus
y(t) = x(s) h(t-s) ds
= h(s) x(t-s) ds

If the input is a complex exponential signal, then

L(e j2vt) = h(s) e j2v(t-s) ds
= h(s) e-j2vsds · e j2vt
= H(v) e j2vt,

i.e. the output also is a complex exponential, multiplied by  H(v)  which is the Fourier transform of the impulse response  h(t).  H  is called the transfer function of the system. Since the output is a convolution of the input and the impulse response,

y(t) = h(s) x(t-s) ds = h(t) * x(t) ,

we have

Y(v) = H(v) X(v) .

The function |H(v)| is called the amplitude response of the system and (v)= arg H(v) is called the phase response of the system. The system is causal , if  h(t) = 0   when   t < 0. Then the values of the input   x(t)   in the future do not affect on the values of the output   y(t)   at the present time instant t and

y(t) = h(s) x(t-s) ds = h(s) x(t-s) ds.

A signal x(t) is causal, if x(t) = 0, when t < 0. If the system is causal and also the input is causal, then the output is a causal signal and

y(t) = h(s) x(t-s) ds.

Example 3.1.1: Smoothing

Example 3.1.2: Transfer function, amplitude response and phase response

Example 3.1.3: Derivator

Example 3.1.4: Transfer function, impulse response and output

Example 3.1.5: Ideal low pass filter

Example 3.1.6: Another ideal low pass filter

In Chapter 4 we give more examples of analog LTI systems.


Problems: P31, P32, T31, P33, P34, P35
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