DSP Notes

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https://www.gaussianwaves.com/2015/11/interpreting-fft-results-complex-dft-frequency-bins-and-fftshift/ << AWESOME

picture of sinc function
Digital means:
  • Discrete time
  • Discrete amplitude

Analogue and digital signals can carry the same information under certain conditions. Answered by Harry Nyquist and Claude Shannon. Under appropriate "slowness" conditions for x(t) we have the Sampling Theorem: x(t) = \sum_{n=-\infty}^{\infty} x[n] \mathrm{sinc}\left(\frac{t - nT_s}{T_s}\right) We can build the continuous time signal from the discrete time sequence. Take copies of the \mathrm{sinc}() function at each sample location scaled by the amplitude of the sample and sum them to get back the original function. The conditions under which you can do this are determined by the Fourier transform. Once we know the "speed" of the signal we can choose a sampling interval that will allow the above theorem to work - this is the Nyquist rate.

Discrete time signal is a sequence of complex numbers denoted x[n] where the square brackets are used to indicate its discrete nature. The index n just provides an ordering for samples which are taken at a steady interval, the sampling period. The sequence is two-sided in that it goes from minus infinity to plus infinity.

The delta signal: \delta[n] . TODO INSERT GRAPH

The unit step signal: \text{u}[n] . TODO INSERT GRAPH

The exponential signal: x[n] = |n|^nu[n], |a| < 1 . TODO INSERT GRAPH. E.g. describes newtons law of cooling or capacitor discharge.

The sinusoidal signal: \sin(\omega_0 n + \theta) . Angles in radians.

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 theoems. 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 include scaling, sum, product. These can be applied to any discrete signal. 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 of signal defined by the following. E_x = \sum_{n = -\infty}^{\infty} |x[n]|^2

Power is the rate of energy production defined as follows. P_x = \lim_{N\to\infty} \frac{1}{2N+1} \sum_{n=-N}^{N} |x[n]|^2


\text{binResolution} = \frac{f_\text{sample}}{\text{# FFT points}} \begin{align} \mathrm{binStartFreq}(n) &= n * \text{binResolution} \\ &= n \frac{f_\text{sample}}{\text{# FFT points}} \end{align}