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Signal Processing & DSP

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

Nyquist Rate

NEWIntermediate
fs2fmaxf_s \ge 2 f_{max}

Sample at least twice the highest frequency to avoid aliasing.

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FFT Bin Frequency

NEWAdvanced
fk=kfsNf_k = \frac{k f_s}{N}

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

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Frequency Resolution

NEWIntermediate
Δf=fsN\Delta f = \frac{f_s}{N}

Spacing between adjacent FFT bins.

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Signal-to-Noise Ratio

NEWBasic
SNR=10log10 ⁣(PsPn)SNR = 10\log_{10}\!\left(\frac{P_s}{P_n}\right)

Ratio of signal power to noise power, in decibels.

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Quantisation SNR

NEWAdvanced
SNR6.02b+1.76SNR \approx 6.02 b + 1.76

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

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Decibel (Power Ratio)

NEWBasic
L=10log10 ⁣(P2P1)L = 10\log_{10}\!\left(\frac{P_2}{P_1}\right)

Express a power ratio in decibels.

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Decibel (Amplitude Ratio)

NEWBasic
L=20log10 ⁣(A2A1)L = 20\log_{10}\!\left(\frac{A_2}{A_1}\right)

Express a voltage/amplitude ratio in decibels.

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RC Filter Cutoff

NEWIntermediate
fc=12πRCf_c = \frac{1}{2\pi RC}

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

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RC Time Constant

NEWBasic
τ=RC\tau = RC

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

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Normalised Sinc Function

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

The ideal interpolation kernel for reconstructing sampled signals.

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Digital Audio Bit Rate

NEWBasic
R=fs×b×cR = f_s \times b \times c

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

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Aliased Frequency

NEWAdvanced
fa=fnfsf_a = |f - n f_s|

Frequency an under-sampled tone masquerades as.

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Discrete Convolution

NEWAdvanced
y[n]=kx[k]h[nk]y[n] = \sum_k x[k] h[n-k]

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

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Decibels to Linear Gain

NEWBasic
G=10L/20G = 10^{L/20}

Convert a decibel gain back to a linear amplitude factor.

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Total Harmonic Distortion

NEWAdvanced
THD=n2Vn2V1THD = \frac{\sqrt{\sum_{n\ge2} V_n^2}}{V_1}

Ratio of harmonic energy to the fundamental.

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