# Examples

## Pass-through system

Consider a discrete-time LTI system with impulse response *h*(*n*)
= *δ* (*n*), the Kronecker delta
function. That is, the impulse response in an impulse, suggesting that the system
does nothing but pass the inputs through to the outputs. The frequency response
is the DTFT of this,

*H*(*ω*) = ∑_{(m
= − ∞
to ∞
)} *h*(*m*)*e ^{−imω}*

= ∑_{(m
= − ∞
to ∞
)} *δ*
(*m*)*e ^{−imω}*

*= e ^{−i}*

^{0}

^{ω}= 1

All terms of the summation are zero except when *m* = 0. The frequency
response also suggests that the system will pass any input through unaltered,
since each complex exponential component of the input will pass through unaltered.

## Delay system

Consider a discrete-time LTI system with impulse response *h*(*n*)
= *δ* (*n *−*
N*), for some integer constant *N*. This system will delay the input
by *N* samples. The frequency response is the DTFT of this,

*H*(*ω*) = ∑_{(m
= − ∞
to ∞
)} *h*(*m*)*e ^{−imω}*

= ∑_{(m
= − ∞
to ∞
)} *δ*
(*m *−* N*)*e ^{−imω}*

=* e*^{−iNω}
.

All terms of the summation are zero except when *m* = *N*. Notice
that the magnitude of this is one,

* | H*(*ω*)
| = 1

but the phase response is a linear function of *ω
*,

∠*H*(*ω*)
= −*Nω*.

A system with such a phase response is called a **linear phase** system.

## Delay system in Continous Time

Consider a continuous-time LTI system with impulse response *h*(*t*)
= *δ* (*t *−*
τ* ), for some real number *τ*.
This system will delay the input by *τ*
seconds. The frequency response is the CTFT of this,

*H*(*ω*) = ∫_{
(− ∞
to ∞
)}*h*(*t*)*e ^{−iω}*

^{t}*dt*

= ∫_{
(− ∞
to ∞
)}*δ*
(*m *−* τ*
)*e ^{−iω}*

^{t}*dt*

= *e*^{−iτ
ω} .

All terms of the integral are zero except when *t* = *τ*
. Notice again that the magnitude of this is one,

* | H*(*ω*)
| = 1

but the the system is again a linear phase system,

∠*H*(*ω*)
= -*τ* *ω.*