EECS20N: Signals and Systems

# 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) = 2π 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) = (1/2π) (− ∞ to ∞ ) X(ω )eiω t

= K (− ∞ to ∞ ) δ (ω ) eiω t

= 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) = 2π K δ (ω )

where δ is the Dirac delta function. This function then periodically repeats with period 2π (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.