### Chirp Synthesis Demo

A linear-FM (chirp) signal is an ideal test signal to explore the concept of aliasing due to sampling. By visualizing the spectrogram of a synthesized chirp and listening to the sound, we experience the fact that a D-to-C converter cannot create output signals with frequencies higher than one half of the sampling frequency. Consult Section 4-2.5 Maximum Reconstructed Frequency for more discussion of this issue. This chirp synthesis demo consists of two simple components (a C-to-D converter and a D-to-C converter) and four steps:
1. Use the mathematical formula for a continuous-time chirp signal $$x(t) = \cos( \pi\alpha t^2 )\qquad 0\leq t \leq T$$ to define a signal whose chirp rate is $$\alpha$$Hz/s, whose duration is $$T$$ seconds.
2. Produce a discrete-time signal by sampling $$x(t)$$ at a rate $$f_s$$ samples/s. This is done by evaluating $$x(t)$$ at $$t=n/f_s$$, and is, in effect, a C-to-D converter. The result is $$x[n] = x(n/f_s)$$.
3. Use Matlab's soundsc function to play the signal for listening. This is the D-to-C conversion component because a continuous-time signal must be created from $$x[n]$$ in order to drive the audio output, i.e., the speakers.
4. Evaluate and display the spectrogram of $$x[n]$$, which gives a visualization of what we hear.
In the demo, the chirp rate is $$\alpha = 1000$$ Hz/s, and the time duration is $$T=3$$s, so the instantaneous frequency $$(f_i(t) = \alpha t)$$ goes from zero at $$t=0$$ to $$3\alpha = 3000$$Hz at $$t=3$$s. Since the highest frequency in $$x(t)$$ is $$3000$$Hz, there should be no aliasing if $$f_s\geq 6000$$Hz.

The demo shows what happens for five different sampling frequencies: 8000, 6000, 4000, 3000, and 2400 Hz. The first two cases exhibit no aliasing and the sound rises from 0 to 3000 Hz. In the last three cases, the sound goes up and down because $$f_s/2<3000$$Hz. When $$f_s=4000$$, the maximum output frequency is $$2000$$Hz, for $$f_s=3000$$, the maximum output frequency is $$1500$$Hz, and for $$f_s=2400$$, the maximum output frequency is $$1200$$Hz.

Spectrograms for the five cases are shown below. Click on the spectrogram to hear the corresponding chirp. Note the different $$y$$-axis scales for frequency. When you listen to the sounds, try to follow the ups and downs that match the spectrograms.