Time-varying signals with multiple frequency components are an important category of signals for engineers. If we press two adjacent keys on a piano simultaneously, we will hear two (approximately sinusoidal) concurrent tones. The resulting sound is probably most readily understood in terms of the individual sinusoids^{1}. We will see later that many engineering design and analysis problems are simplified by examining the response of the system based on the sinusoidal frequencies contained in the inputs applied to the system.
Sinusoidal sweeps are a relatively simple variation of the sinusoids we saw in the first experiment of this module. A sinusoidal sweep is a sinusoid whose frequency varies with time. (Thus, the peaks in the sinusoid will get closer together or further apart as time goes on.) An example of a sinusoidal sweep is shown in Fig. 1. Sinusoidal sweeps have practical applications in the testing of engineering systems. By applying a sinusoidal sweep to a system, we can measure the system's response to sinusoids of various frequencies. These measured responses can be used to predict the system's response to other inputs. For now, we will be using sinusoidal sweeps to provide some fundamental insight into the topic of time-varying signals with multiple frequency components.
In the remainder of this document, we present the definitions and details associated with the mathematics of basic sinusoidal sweep signals.
Sinusoidal sweeps^{2} are functions of the following form:
(1.)
\[f(t) = A\cos (\omega (t) \cdot t)\]Where the instantaneous frequency, ω(t), of the sinusoid is a time-varying function. As in our previous experiment, A is the amplitude of the sinusoidal sweep.
(2.)
\[\omega (t) = {\omega _i} + a \cdot t\]Where ω_{i} is the initial frequency of the swept sinusoid, and a defines the rate of change of the frequency. We will define a as follows:
(3.)
\[a = \frac{{{\omega _f} - {\omega _i}}}{{{T_T}}}\]Where ω_{f} is the final frequency of the swept sinusoid, and T_{T} is the total duration of the signal.
Thus, the swept sinusoid is completely defined by the following parameters:
Units of Hertz (Hz) can also be used to specify the initial and final frequencies. Just be sure that you are aware of what system of units you are using and interpret your results accordingly.
A typical swept-sinusoidal signal is shown in Fig. 1. Notice that the period (and thus the frequency) of the signal in Fig. 1 varies as time varies. The period at the beginning of the signal is T_{i }, which is approximately \(\frac{{2\pi }}{{{\omega _i}}}\), and the period at the end of the signal is T_{f}, which is approximately \(\frac{{2\pi }}{{{\omega _i}}}\). Since the period is decreasing over the duration of the signal, the frequency is increasing.