The Role of Complex Sinusoids in System Characterization

Complex sinusoids are disproportionately emphasized among all signals in DSP formulations. Why? This is what we are going to investigate now.

System input

Let’s probe a system that has an impulse response $h[n]$ with a complex sinusoid $x[n]=e^{j\omega_0 n}$ as shown in the figure below:

• Keep in mind that the frequency response is the Fourier transform of the impulse response $h[n]$.

$$H(\omega)=\sum_n h[n]e^{-j\omega n}$$

• The system output is given by the convolution between the input $x[n]=e^{j\omega_0 n}$ and the impulse response $h[n]$.

$$y[n]=\sum_m x[m]h[n-m]$$

In such a simple setup, we plug in the value of $x[n]$ and see where it leads.

$$\begin{align*} y[n]&=\sum_m e^{j\omega_0 m}h[n-m]\\ &= \sum_m e^{j\omega_0 (m-n+n)}h[n-m]\\ &= e^{j\omega_0 n}\sum_m h[n-m]e^{-j\omega_0(n- m)}\\ &= e^{j\omega_0 n}H(\omega_0) \end{align*}$$ ...