Frequency-Domain Representation and Ideal Signals

• When working with signals and systems, it is generally useful to consider both time- and frequency-domain representations.

• The frequency-domain representation of a signal indicates how that signal can be constructed in terms of sinusoidal components, each with a specific frequency, gain and phase offset.

• We will often make use of “ideal” signals, such as sinusoids, noise and impulses, as inputs to our systems.

• Sinusoids: can be represented mathematically in the time-domain with $\sin$ or $\cos$ functions; in the frequency domain, they consist of a single non-zero frequency component.

• Noise: in the time domain, noise signals consist of a continuous stream of randam values; in the frequency domain, they contain sinusoidal energy over a specified wide range of frequencies.

• Impulses: in the time domain, a signal of very short time duration and an area of one; in the frequency-domain, impulses contains sinusoidal energy over a full range of frequencies.

• A discrete-time unit impulse signal is defined in the time domain by:
  $$\delta[n] \ensuremath{\stackrel{\Delta}{=}}\left\{ \begin{array}{ll} 1, & n = 0 \\ 0, & n \neq 0 \end{array} \right.$$

• The frequency-domain representation of a discrete-time signal is computed using the discrete fourier transform (DFT). The resulting frequency-domain signal is given in terms of complex numbers of the form $a + j b$, which can be plotted in terms of magnitude and phase values. Magnitude values are calculated as $\sqrt{a^2 + b^2}$, while phase values are calculated as $\tan^{-1}(b/a)$.

• Figure 1 illustrates example sinusoidal, noise and impulse signals, plotted in both the time and frequency domains (magnitude only).

  Figure 1: Sinusoidal (top), noise (middle) and impulse (lower) signals: time- and frequency-domain representations.