f(t) is the input and g(t) is the output.
f(t) -> g(t)
f(t+) -> g(t+) time invariant
f_{1}(t) -> g_{1}(t) | => af_{1}(t) + bf_{2}(t) -> ag_{1}(t) + bg_{2}(t) linear |
f_{2 }(t) -> g_{2 }(t) |
Let h be the impulse response of the system,
We can write
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) | |
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^{-j2vs}ds · 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,
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
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
Example 3.1.2: Transfer function, amplitude response and phase response
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.