There are four sounds to study. First listen to each of them without
knowing what they are.### Plot Linearly Varying Chirp in the Time Domain

Let's look at one of them plotted in the time domain (i.e., \(x(t)\) vs. \(t\)).
The time signal for a long LFM chirp is viewed by plotting the signal
along successive rows. Each row covers 60 milliseconds, so only the
first quarter of the signal is shown.
### Time-frequency domain

Another way to study the sounds is to view them in the
time-frequency domain.
Listen to each of the signals again by clicking on the waveform.### Instantaneous Frequency

Did you notice that all of spectra changed with time? The movement of
frequency, however, is simple enough that you can describe it with a
simple mathematical equation such as a straight line or a sine wave.
The concept of instantaneous frequency provides the correct
way to derive the frequency versus time behavior for a chirp.
### Comparison of the plot of the chirp to constant
frequency sinusoids

The following example shows a
comparison of the plot of the chirp to constant
frequency sinusoids for reference.
One way to understand the concept of instantaneous frequency is to compare
the plot of a LFM chirp side-by-side with the plot of a constant frequency
cosine wave. In the plot below the chirp is placed between a 300 Hz
sinusoid on top and a 500 Hz sinusoid on the bottom.
** instantaneous frequency ** rises
linearly at a rate of 200 Hz per 20 milliseconds.
As a result, the formula for instantaneous frequency versus time must be

Did you notice that they all had a distinct movement of frequency vs. time.

Notice how the period is getting shorter for later times.

Count the number of periods in two regions:

- for t between 0 and 10 milliseconds (the beginning of the first row)
- for t between 240 and 250 milliseconds (the beginning of the last row)
- Estimate the frequency in these two regions by taking 1/Period. Did you get 200 Hz and 500 Hz?

Can you point to features in the spectrograms that correspond to the changing tones (or pitch) in the sounds your hear?

**Challenge:** write the mathematical formula for the sine-wave
modulated chirp. Use the concept of instantaneous frequency.

How would you model the bird songs with mathematically generated synthetic chirps?

These were generated with MATLAB.
Click here to see the code that computes and
displays the spectrograms.

This is done by breaking the signal into many parts and computing the spectrum of each part.
The spectra are plotted with a dark component representing a strong
spectral component and white meaning no energy.

Listen to the chirp with a linear frequency movement versus time .

This sound is synthesized via the formula:

$$x(t) = \cos(2\pi(m t+f)t )$$

The frequency that will be heard is determined by taking the derivative of the quantity \(2\pi(m t+f)t\) which is the argument of the cosine. If we start with \( \cos(P(t))\), the derivative must be divided by \(2\pi\) to get the frequency in Hertz.

$$f_i(t) = \frac1{2\pi} \frac{dP(t)}{dt}$$

Therefore, if \(m=5,000\) and \(f=100\), the frequency can be calculated to be $$f_i(t) = 2m t + f = 10,000 t + 100 \quad\text{(hertz)}$$

Where do the frequencies line up?

- the top plot seems to match near t = 20 milliseconds (0.02 secs).
- the bottom plot matches near t = 40 milliseconds (0.04 secs).

$$f_i(t) = 10,000 t + 100 (Hertz)$$