# Constant Signals

## Continuous Time

Consider a continous-time signal

*x*(*t*) = *K*,

for some real constant *K*. Its CTFT is

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

*dt*

which is not easy to evaluate. This integral is mathematically very subtle. The answer is

∀*ω
*∈*
Reals*, *X*(*ω*)
= 2*π* *K δ
*(*ω* )

where *δ * is the Dirac delta function.
What this says is that a constant in the time domain is concentrated at zero
frequency in the frequency domain (which should not be surprising). We can verify
this answer easily by evaluating the inverse CTFT,

*x*(*t*) = (1/2*π*)
∫_{
(− ∞
to ∞
)}* **X*(*ω*
)*e ^{iω t}*

*dω*

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

*dω*

*= K,*

where the final step follows from the **sifting property** of the Dirac
delta function. (To see how the sifting property works, note that the integrand
is zero everywhere except where *t* = 0, at which point the complex exponential
evaluates to 1. Then the integral of the delta function itself has value one.)

## Discrete Time

Consider a discrete-time signal

*x*(*n*) = *K*,

for some real constant *K*. Its DTFT is

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

which is not easy to evaluate. This sum is mathematically very subtle. The answer is

∀*ω
*∈*
*[-*π*,*π*], *X*(*ω*)
= 2*π* *K δ
*(*ω* )

where *δ * is the Dirac delta function.
This function then periodically repeats with period 2*π*
(as it must to be a DTFT). What this says is that a constant in the time domain
is concentrated at zero frequency in the frequency domain (which should not
be surprising). We can verify this answer easily by evaluating the inverse DTFT.