Spectrum of Sampled Complex Sinusoid

In the discrete-time case, we replace $t$ by $nT$ where $n$ ranges over the integers and $T$ is the sampling period in seconds. Thus, for the positive-frequency component of the sinusoid of the previous section, we obtain

$$s_{\omega_0}(n) \isdef e^{j\omega_0 n T}.$$

It is common notational practice in signal processing to use normalized radian frequency

$${\tilde \omega}\isdef \omega T \;\in[-\pi,\pi).$$

Thus, our sampled complex sinusoid becomes

$$s_{\omega_0}(n) \isdef e^{j{\tilde \omega}_0 n}.$$

It is not difficult to convert between normalized and unnormalized frequency. The use of a tilde (` $\tilde{\null}$') will explicitly indicate normalization, but it may be left off as well, so that $\omega$ may denote either normalized or unnormalized frequency.^5.4

The spectrum of infinitely long discrete-time signals is given by the Discrete Time Fourier Transform (DTFT) (discussed in §2.1):

$$S_{\omega_0}(\omega) \isdef \sum_{n=-\infty}^{\infty} s_{\omega_0}(n) e^{-j\omega n} = 2\pi\delta(\omega-\omega_0) = \delta(f-f_0)$$

where now $\delta(\omega)$ is an impulse defined for $\omega\in[-\pi,\pi)$ or $f\in\left[-\frac{1}{2},\frac{1}{2}\right)$, and $\omega$ denotes normalized radian frequency. (Treatments of the DTFT invariably use normalized frequency.)