3.11 The sinusoidal steady state response

It is useful to see what the effect of the filter is on a sinusoidal signal, say $x\left(t\right)=cos\left({\Omega }_{0}t\right)$ . If $y\left(t\right)$ is the output of the filter, then we can write

Using the Euler formula for $cos\left({\Omega }_{0}t\right)$ , right hand side of [link] can be written as:

This integral can be split into two separate integrals, and written as:

The first of the two integrals can be recognizes as the Fourier Transform of the impulse response evaluated at $\Omega ={\Omega }_0$ . The second integral is just the complex conjugate of the first integral. Therefore [link] can be written as:

Since the second term in [link] is the complex conjugate of the first term, we can express [link] as:

or expressing $H\left(j{\Omega }_0\right)$ in terms of polar coordinates:

Therefore, we find that the filter output is given by

This is called the sinusoidal steady state response . It tells us that when the input to a linear, time-invariant filter is a cosine, the filter output is a cosine whose amplitude has been scaled by $\left|H\right(j{\Omega }_0\left)\right|$ and that has been phase shifted by $\angle H\left(j{\Omega }_0\right)$ . The same result applies to an input that is an arbitrarily phase shifted cosine (e.g. a sine wave).

Example 3.1 Find the output of a filter whose impulse response is $h\left(t\right)=e^{-5t}u\left(t\right)$ and whose input is given by $x\left(t\right)=cos\left(2t\right)$ . It can be readily seen that the frequency response of the filter is

and therefore $\left|H,\left(,j,2,\right)\right|=0.1857$ and $\angle H\left(j2\right)=-0.3805$ . Therefore, using [link] :