**Fourier Analysis**

- Any reasonably behaved periodic function with period T can be constructed as the sum of a series of sines and cosines

- where 1/T is the fundamental frequency and a and b represent the coefficients to the sinusoids
- we can then compute the coefficients using the following methods in particular this particular derivation takes advantage of integrating to boundary conditions that only sin and cos can achieve namely nonzero at x = 0 to be a cosine and 0 at x = 0 to be a sin, and the 0 nonzero capabilities of the functions at multiples of pi and pi/2

- by using the above knowledge we can choose to solve for those points where sin/cosine must be 0 or nonzero by knowing that one of the constants cannot be or must be 0. This allows us to set one constant to 0.
- for example, if we integrate sin in the above from 0 to T like is shown above, then multiply both sides of the equation by sin(2πkft) we get can solve for a
_{n}to be what is shown below, since all the b_{n}terms will cancel out due to integrating a cos from 0 through one period - this is the same as what can be done for cos, which gives us the b
_{n}terms

**Bandwidth-Limited Signals**

- transmission of ASCII characters
- b encoded is 01100010

- giving us this fourier analysis result
- we can see this more clearly in the following diagram

- so as we can see fourier analysis is the same as finding the shadow of a waveform
- the same as how we can decompose a vector into i j k components, we can decompose a sinusoid into the sum of base sinusoids with different coefficients
- width of the frequency range we can transmit through is called the bandwidth
- signals that run from 0 to a maximum frequency are called baseband signals
- signals that are shifted up to a higher range of frequency are called passband signals
- all wireless signals are passband
- Voice-grade line
- artificially induced cutoff just above 3000 Hz
- this is used for ordinary telephone lines
- copper wire
- to EE majors(analog)
- bandwidth is measured in Hz
- to Computer Science Majors(digital)
- bandwidth is measured in bits/sec for transmission

**The Maximum Data Rate of a Channel**

- Henry Nyquist realized that even perfect channels have finite transmission capabilities
- Shannon progressed this work
- Nyquist's Theorem

- So for example a noiseless 3kHz channel that transmits binary, two level signals would have a max transmit rate of
- 6000 b/s
- Shannon's include the idea of SNR(Signal to Noise Ratio) for noisy channels
- SNR is usually measured in decibels, where dB = 10 log
_{10}S/N

- which is a closer approximation for real channels

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