Lectures: Signal Processing

🎓 Theory

Before computing, decide whether the signal is deterministic or random, continuous or discrete, and whether the system is linear and invariant. If it is, the response is a convolution and the frequency response describes the system completely (Bracewell, chapters 1 and 9).

❓ Frequently asked

📌 Worked example with real numbers

Feed a 50 Hz sinusoid into an amplifier. If the output has an extra 100 Hz line, the system is nonlinear (module 1).

🎓 Theory

To keep a signal you must sample at f_s above twice its highest frequency. A component above f_s/2 is folded onto a lower frequency and cannot be separated after sampling (Bracewell, chapter 10; Barkat, section 3.8).

• f_ssampling rate

❓ Frequently asked

📌 Worked example with real numbers

A 5000 Hz wave sampled at f_s=8000 Hz (Nyquist 4000 Hz). The FFT peaks at 3000 Hz, exactly |5000-8000|.

🎓 Theory

Observing for time T resolves only lines at least 1/T apart. Cutting the signal with a rectangular window leaks the spectrum into neighbouring bins (module 1); this is the convolution with a sinc function seen from the other side.

❓ Frequently asked

📌 Worked example with real numbers

With f_s=8000 Hz and 1024 samples the bin spacing is 7.8125 Hz. To separate two lines 5 Hz apart you need T\ge0.2 s, i.e. at least 1600 samples.

🎓 Theory

Pick a convention (where the 2\pi sits) and exploit symmetry: a real signal has a Hermitian spectrum, so half of it suffices. The shift, similarity, derivative and convolution theorems (chapter 6) find transforms without integrating (chapter 7).

❓ Frequently asked

📌 Worked example with real numbers

A real 1000-sample signal has 1000 complex FFT coefficients, but thanks to the Hermitian property only 501 are needed (rfft).

🎓 Theory

A filter is a convolution with the impulse response and, in the frequency domain, a multiplication by the frequency response. An FIR filter (running mean) is always stable; a recursive one such as y[n]=a\,y[n-1]+(1-a)x[n] is cheaper but stability must be checked (modules 1, 3 and 9).

❓ Frequently asked

📌 Worked example with real numbers

An 8-point running mean at f_s=8000 Hz has gain 0.640729 at 500 Hz and 0 at 1000 Hz (a null). Both freqz and the closed form agree.

🎓 Theory

The density of a sum of independent random variables is the convolution of the densities, i.e. the characteristic functions multiply. The same convolution, the same theorem (Bracewell, chapter 16; Barkat, chapters 1 and 2).

❓ Frequently asked

📌 Worked example with real numbers

The sum of two independent variables uniform on [-\tfrac12,\tfrac12] has a triangular density. Numerical convolution gives variance 0.16667, matching 1/12+1/12=1/6.

🎓 Theory

The power spectrum is the Fourier transform of the autocorrelation. The periodogram of one record has a standard deviation about equal to its own value and does not shrink as N grows, so several records must be averaged (Bracewell, chapters 17 and 19; Barkat, sections 3.5 and 3.7).

• R(\tau)autocorrelation function
• S(s)power spectrum

❓ Frequently asked

📌 Worked example with real numbers

Unit white noise, 1024 samples per record: one periodogram has standard deviation 1.02 (theory 1), averaging 64 records leaves 0.125 (theory 0.125).

🎓 Theory

Detection chooses between H_0 (noise only) and H_1 (signal present). The Neyman–Pearson criterion fixes the false-alarm probability P_{FA} and maximises the detection probability P_D; the solution is a likelihood-ratio test (Barkat, chapters 5 and 10).

• Qupper-tail function of the normal distribution

❓ Frequently asked

📌 Worked example with real numbers

Detecting a DC level A=1 in Gaussian noise \sigma=1 with N=10 samples and P_{FA}=10^{-3}: the threshold on the sample mean is 0.9772, d=3.1623, P_D=0.5287 by formula and 0.528 by a 400 000-trial simulation.

🎓 Theory

Maximum-likelihood (ML) estimation picks the parameter that makes the observed data most likely; Bayes estimation adds a prior (MAP, MMSE). The Cramér–Rao bound gives the smallest variance of any unbiased estimator (Barkat, chapter 6).

❓ Frequently asked

📌 Worked example with real numbers

Estimating a DC level from N=10 samples of noise \sigma=1: the Cramér–Rao bound is 0.1, the simulated variance of the sample mean is 0.1001, so the sample mean is an efficient estimator.

🎓 Theory

The orthogonality principle says the error of a least-squares estimate must be orthogonal to the data. The Wiener filter applies it to stationary processes; the Kalman filter works recursively for time-varying systems (Barkat, sections 7.2, 7.3 and 7.5).

❓ Frequently asked

📌 Worked example with real numbers

A scalar signal of variance 1 in noise of variance 0.25: Wiener gain 0.8, mean-square error 0.2 (simulated 0.2006) versus 0.2504 if the data is left alone.

🎓 Theory

When the noise power varies, a fixed threshold lets the false-alarm probability drift. CFAR estimates the noise level from reference cells around the cell under test and sets a proportional threshold, keeping P_{FA} constant (Barkat, chapters 11 and 12).

❓ Frequently asked

📌 Worked example with real numbers

CA-CFAR with N=16 reference cells and P_{FA}=10^{-2} in square-law noise: threshold factor \alpha=5.336 times the reference-cell mean; a 500 000-trial simulation gives P_{FA}=0.0099.

🎓 Theory

A result is not finished until two independent methods agree: FFT with a direct sum, numerical integral with a closed form, formula with simulation. When reporting, state f_s, N, the window, the random seed and the library versions so others can reproduce it.

❓ Frequently asked

📌 Worked example with real numbers

Every notebook in this course does exactly this: for example module 1 prints the deviation between FFT and direct sum and stops if it exceeds 10^{-9}.