Basic DSP methods

Equation for Sampling Theorem

Fs​ ≥ 2fmax​

Where Fs is sampling frequency, fmax is the maximum frequency

Equation for Sampling Theorem in practice

Fs​ ≈ 10fmax​

Where Fs is sampling frequency, fmax is the maximum frequency

Equation for Aliasing Prevention

fc ​≤ Fs/2

Where Fs is sampling frequency, fc is the cutoff frequency,

Explain aliasing

occurs when a signal is sampled below twice its maximum frequency, causing higher frequency components to appear as lower frequencies.

How to prevent aliasing?

using an analogue low-pass filter with cut-off ≤ Fs/2.

Equation for number of levels (in quantisation)

N = 2^b

Where N is the number of levels, b is the bit depth(resolution) - determines the

number of discrete values used

to describe the analogue signal

Equation for quantisation step

q = Dynamic Range/(2^b)

Where q is the maximum amplitude, b is the bit depth

What is the relationship between quantisation noise and bit depth

Quantisation noise decreases as bit depth increases and as the signal better utilises the ADC dynamic range.

What bit depth is needed for SNR = 256 assuming full dynamic range?

SNR=2b⇒b=log₂(256)=8

Equation for mean, supposing we have a discrete valued signal 𝑧(𝑘). For the interval 0 to

N

µ = (1/N)Σ(z(k))

Equation for variance, supposing we have a discrete valued signal 𝑧(𝑘). For the interval 0 to

N

σ² = (1/N)Σ(z(k) - µ)²

Equation for Gaussian PDF, supposing we have a discrete valued signal 𝑧(𝑘). For the interval 0 to

N

p(n) = (1/√(2πσ))e^-((n-µ)²/2σ²)

How to best describe Random Noise

statistically and typically follows a Gaussian distribution characterised by its mean and standard deviation.

How to calculate SNR estimate

SNR = Signal/σ

Formula for SNR Averaging

𝑆𝑁𝑅𝑎𝑣𝑔 = √𝑁 ∙ 𝑆𝑁𝑅

How does Averaging improve SNR

by reducing random noise while preserving the coherent signal.

State the conditions under which averaging is effective

Noise is random

Noise has zero mean

Signal is constant

Signal and noise are uncorrelated

What is correlated noise?

coherent, repeatable interference that appears consistently across measurements, making it resistant to averaging but removable through differential processing.

Examples of Correlated Noise

mains hum, power supply ripple, interference (from external sources)

What is differential processing

removes coherent noise by estimating the common component and subtracting it from each measurement.

DIfferential Processing Equation

s_clean = s - s_common

Where s is the measured signal(contains real signal and the correlated noise), s_common is the estimated common noise, s_clean is the real signal (final cleaned signal)