# DSP Notes

## Page Contents

## References

None

## To Read

https://www.maths.ox.ac.uk/system/files/attachments/complex%20%281%29.pdf https://ocw.mit.edu/courses/mathematics/18-03sc-differential-equations-fall-2011/unit-i-first-order-differential-equations/complex-arithmetic-and-exponentials/MIT18_03SCF11_s6_5text.pdf https://www.pearsonschoolsandfecolleges.co.uk/Secondary/Mathematics/16plus/EdexcelModularMathematicsforASandALevel/Resources/FurtherPureMathematics2/03%20Ch%2003_018-065.pdf http://faculty.uml.edu/cbyrne/SP1text.pdf http://www-math.mit.edu/~gs/papers/newsigproc.pdf http://www.ling.upenn.edu/courses/ling525/Moore1978Part1.pdf http://www.ling.upenn.edu/courses/ling525/Moore1978Part2.pdf http://greenteapress.com/thinkdsp/thinkdsp.pdf http://web.eecs.umich.edu/~aey/eecs206/lectures/phasor.pdf https://www.gaussianwaves.com/2015/11/interpreting-fft-results-complex-dft-frequency-bins-and-fftshift/ << AWESOME https://math.stackexchange.com/questions/9416/extracting-exact-frequencies-from-fft-output https://dsp.stackexchange.com/questions/38577/fft-starting-at-zero

## An Intro To Digital Signals

### What Is A Digital Signal?

- Discrete time
- Discrete amplitude

Analogue and digital signals can carry the same information under certain conditions. Answered by
Harry Nyquist and Claude Shannon. When the sampling frequency is at least double the maximum frequency in the signal being sampled, the sampling theoroem holds:
*Nyquist* rate.

Discrete time signal is a sequence of *complex* numbers denoted

The delta signal:

The unit step signal:

The exponential signal:

The sinusoidal signal:

There are 4 signal classes:

- finite length - only N samples. Range of index is 0 to N-1.
- infinite length - index N ranges over entire range of integers. abstract. good for theorems. They have
*infinite*energy. - periodic - data repeats every N samples. Represent with a tild on top. Same info as a finite-length of length N. They have
*infinite*energy. - finite support - infinite length with on a finite number of non zero samples. Eg unit step.

### Elementary Operations On Digital Signals

Elementary operations include scaling, sum, product and shift. These can be applied to any discrete signal.

- Scale:
.$y[n]=\alpha x[n],\text{}\alpha \in \mathbb{C}$ - Sum:
.$y[n]=x[n]+z[n]$ - Product:
.$y[n]=x[n]\cdot z[n]$ - Shift:
.$y[n]=x[n-k],\text{}k\in \mathbb{Z}$

Shift-by-k can be applied to infinite signals, but when applying to discrete time signals we need
to state what happens when we go beyond the index range N, i.e., how we embed it into an
infinite length sequence. Can embed into a finite-support sequence by putting zeros on the
left and right. Or we could use a periodic extension, whereby the shift becomes circular.
*The periodic extension is the natural way to interpret the shift of a finite-length signal*.

### Energy And Power

Energy of signal defined by the following.

Power is the rate of energy production defined as follows.

### Analog To Digital And Back Again

Analog signals are sampled at regular intervals to produce digital signals. The time between samples is called the *sampling period* and is detoted

If

There is thus the following relationship between discrete samples and anlalog time:

Therefore, if a signal has a period of

For example, a typical sampling frequency for audio is

In this case, if we know that there are 110 samples in the signal's period, i.e.

### Discrete Time Sinusoids Not Always Periodic

A discrete complex exponential is only period if the frequency is a rational number (any number that's a fraction

I.e., to be periodic the frequency must satisfy:

The *fundamental frequency* is the smallest value of N for which

### Aliasing

Discrete time sinusoids whos frequencies are separated by an integer multiple of

This gives rise to something called *aliasing*, where any discrete time sinusoid has
multiple equivalends, or aliases, separated by integer multiples of

## DFT/FFT

Where has 3Blue1Brown been all my life?!

http://www.tedknowlton.com/resume/FFT_Bin_Interp.html https://electronics.stackexchange.com/questions/12407/what-is-the-relation-between-fft-length-and-frequency-resolution