# The relation between a link speed and Fourier coefficients

I am reading Computer Networks by Tanenbaum and the the chapter about the physical link and there is an example that I don't understand.

The author states that given a bit rate of b bits/sec the time required to send 8 bits is 8/b bits/sec. thus the frequency of the first harmonic is 8/b. Then there is an example of a telephone line which have a cutoff at 3000 Hz and claims that the highest harmonic passed is 3000 /(b/8) = 24000/b.

An Example is given for a 300 Bps, the transmission time is 26.67 msec, the first harmonic is 37.5 Hz and the number of harmonics sent is 80.

I understand that a real signal is finite and we can regard it as having a period of some T. I noted that the first harmonic is at 1/T.

I have a few questions:

1. When sending a stream of data, do we do modulation for some of the bits up to a point and this is the signal we calculate its period etc' ? we can't read all the data to transmit before transmission.

2. I get that we can reconstruct the signal from the Fourier coefficients. I didn't understand the relation between harmonics and coefficients (does sending k harmonics means sending k coefficients ?)

3. I don't understand the calculation that was made to get that we can send 80 harmonics in the above example, why does it taking the bandwidth and dividing it to the frequency of the first harmonic ?

4. Is it possible or wished for that the harmonics sent would not be the first harmonic and all of its multiples up to a certain point ? e.g sending the first harmonic and sending the third one but not the second one.

I'd appreciate an answer to any of those questions, I have studied Fourier analysis but with no relation to signals and real world applications so I'm having difficulties putting theory to use.

## 1 Answer

I'll try to group all your questions together because they are related.

To find the spectrum of a signal (frequency content) one has to look at a finite period of time. Let's say one looks at the 1st 10 seconds of a signal. The fundamental frequency is the sinusoidal signal that can make 1 revolution in the measured time. The period is 10 seconds, the fundamental frequency is 0.1 Hz. If one is looking at the discrete Fourier transform of a signal they are looking to see how many multiples of this fundamental frequency there are. The 1st frequency is 0.1 Hz, the 2nd is 0.2 Hz, the 13 is 1.3 Hz, etc. The coefficients are the amount of each frequency present (the 0.5 Hz signal is the 5th harmonic of the fundamental frequency, which is 0.1 Hz). One could say they have 0.3 of the 1st harmonic, 0.2 of the 8th harmonic, etc. Different signals are made by having different amounts of each harmonic.

Certain signals might not have all harmonics. Ideal square waves are made of odd harmonics (0.1 Hz, 0.3 Hz, 0.5 Hz, etc, but no 0.2 Hz, 0.4 Hz, etc). Sawtooth waves are made of only even harmonics.

When using the DFT there is a midpoint where signals start aliasing. If you have 100 samples of your signal over 10 seconds then the highest frequency that should be present in your original signal is a 5 [Hz] signal. This is due to aliasing where 4.9 Hz looks like 5.1 Hz, 0.1 Hz looks like 9.9 Hz, etc.

Short answers to your questions:

1. Period is calculated based on the signal used to calculate the Fourier transform. You can rx/tx whatever you want.

2. Coefficients represent the quantity of each harmonic.

3. Bandwidth limits the frequencies that can be used. Cutoff = 10 Hz, only Dc through 10 Hz signals can be sent. 1st harmonic is calculated from the period of the signal used for Fourier transform.

4. Depends on the signal being sent, not all signals have all frequencies.