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Digital signal processing analyses and modifies signals represented as sequences of numbers, after sampling and quantisation. It covers sampling theory, convolution, transforms such as the DFT and z-transform, and the design of digital filters for audio, communications, control and instrumentation.

The sampling theorem states that a signal can be reconstructed exactly only if it is sampled at more than twice its highest frequency component. That threshold is the Nyquist rate. Sampling below it causes aliasing, which cannot be corrected once the samples are taken.

Aliasing occurs when a signal is sampled too slowly, so frequencies above the Nyquist frequency appear as lower frequencies that were never present. The false components are indistinguishable from real ones afterwards, which is why an anti-aliasing filter is applied before sampling.

Because aliasing happens during sampling itself. Once a high frequency has folded down to a lower one, the two are identical in the sampled data, and no later filter can separate them. Removing the high frequencies beforehand prevents the folding from happening.

Convolution combines an input sequence with a system's impulse response to give the output. Each output sample is a sum of products of input samples and reversed impulse response samples. For sequences of lengths M and N, the result has length M plus N minus one.

None mathematically: the fast Fourier transform is an efficient algorithm for computing the discrete Fourier transform. A direct DFT costs on the order of N squared operations while the FFT costs on the order of N log N, which matters enormously for large transforms.

Bin spacing equals the sampling rate divided by the number of samples. More samples give finer resolution; a higher sampling rate with the same number of samples gives coarser resolution. Zero padding smooths the display but does not add genuine resolution.

Spectral leakage is the spreading of a tone's energy into neighbouring frequency bins, caused by analysing a finite block of samples. The implicit rectangular window has high side lobes, so tapered windows are used to reduce leakage at the cost of a wider main lobe.

A window function tapers the ends of a sample block to reduce spectral leakage. Hamming, Hanning and Blackman windows trade side lobe suppression against main lobe width, so the choice depends on whether separating close tones or suppressing leakage matters more.

The z-transform converts a difference equation into an algebraic expression in the complex variable z, giving a transfer function. Its poles and zeros reveal the system's frequency response and stability, so it plays the role that the Laplace transform plays for continuous systems.

A causal discrete-time system is stable when every pole of its transfer function lies strictly inside the unit circle in the z-plane. A pole exactly on the circle gives a response that does not decay, and any pole outside it makes the output grow without bound.

FIR filters have no feedback, so they are always stable and can give exactly linear phase, but need a higher order for a sharp response. IIR filters use feedback, achieving sharp responses at low order, but stability must be checked and linear phase is not generally available.

Use an FIR filter when phase must be preserved, as in audio processing and data transmission where phase distortion would smear the signal, or when guaranteed stability matters. The cost is a higher filter order and so more computation per output sample.

Usually by designing an analogue prototype such as Butterworth, Chebyshev or elliptic to meet the specification, then mapping it to the discrete domain with the bilinear transform, applying frequency pre-warping so the critical frequencies land where intended after the mapping.

Yes. A one-to-one tutor can work through convolution, DFT, z-transform and filter design problems with you, explain stability and the FIR against IIR choice, and check your MATLAB work. You complete your own assessed work in line with your course rules.