Signal Processing & DSP

Turning signals into data: sampling, Nyquist, FFT bins, SNR, filters and quantisation

Signal Processing & DSP formulas help you turn real-world signals into usable data, from sampling and Nyquist limits to FFT bins, filters, quantisation, and decibels. This category is useful for students, exam revision, and engineers working with audio, communications, and electronics, because these equations explain how signals are captured, analysed, distorted, and improved in practical systems.

Nyquist Rate

NEWIntermediate

f_s \ge 2 f_{max}

Sample at least twice the highest frequency to avoid aliasing.

FFT Bin Frequency

f_k = \frac{k f_s}{N}

Frequency represented by the k-th bin of an N-point FFT.

Frequency Resolution

NEWIntermediate

\Delta f = \frac{f_s}{N}

Spacing between adjacent FFT bins.

Signal-to-Noise Ratio

SNR = 10\log_{10}\!\left(\frac{P_s}{P_n}\right)

Ratio of signal power to noise power, in decibels.

Quantisation SNR

SNR \approx 6.02 b + 1.76

Best possible SNR (dB) of an ideal b-bit converter.

Decibel (Power Ratio)

L = 10\log_{10}\!\left(\frac{P_2}{P_1}\right)

Express a power ratio in decibels.

Decibel (Amplitude Ratio)

L = 20\log_{10}\!\left(\frac{A_2}{A_1}\right)

Express a voltage/amplitude ratio in decibels.

RC Filter Cutoff

NEWIntermediate

f_c = \frac{1}{2\pi RC}

Corner frequency of a simple RC low/high-pass filter.

RC Time Constant

\tau = RC

Time for an RC circuit to charge to ~63% of final value.

Normalised Sinc Function

\text{sinc}(x) = \frac{\sin(\pi x)}{\pi x}

The ideal interpolation kernel for reconstructing sampled signals.

Digital Audio Bit Rate

R = f_s \times b \times c

Raw data rate from sample rate, bit depth and channels.

Aliased Frequency

f_a = |f - n f_s|

Frequency an under-sampled tone masquerades as.

Discrete Convolution

y[n] = \sum_k x[k] h[n-k]

Output of a filter is the convolution of input and impulse response.

Decibels to Linear Gain

G = 10^{L/20}

Convert a decibel gain back to a linear amplitude factor.

Total Harmonic Distortion

THD = \frac{\sqrt{\sum_{n\ge2} V_n^2}}{V_1}

Ratio of harmonic energy to the fundamental.