Real-World Mathematics: Signal Processing, Sampling, Filtering, Noise and Fourier Analysis

Application of Mathematics in Real-World Usage · Guide 31 · BTT Mathematics Hub

A microphone voltage, temperature record, vibration trace or radio waveform can all be represented as signals: quantities that vary with time, position or another independent variable. Signal processing asks what structure is present, what information sampling preserves, what filtering removes, and how noise changes what can be inferred.

This guide uses fictional signals and ideal mathematical models. It is not communications-system design, medical signal interpretation or safety-critical measurement guidance. Real systems require calibrated sensors, anti-alias filtering, uncertainty analysis and application-specific validation.

A sinusoid separates amplitude, frequency and phase

A common model is x(t)=A cos(2πft+φ). A is amplitude, f is frequency in cycles per second and φ is phase.

For f=5 Hz, period T=1/f=0.2 s. Changing amplitude changes vertical scale; changing frequency changes repetition rate; changing phase shifts the waveform in time.

Sampling converts continuous time into a sequence

At sampling frequency fs, samples occur every Ts=1/fs seconds. If fs=1,000 samples/s, sample interval is0.001s or1ms.

A two-second record at 1,000 samples/s contains 2,000 samples if the counting convention is duration multiplied by rate without double-counting both endpoints.

Sampling must be fast enough for the represented bandwidth

The ideal bandlimited sampling theorem says a continuous-time signal with no content at or above half the sampling frequency can be reconstructed under its assumptions. In practical teaching language, fs must exceed twice the highest represented frequency.

A bandlimited signal containing frequencies only up to400Hz therefore needs a sampling rate above800Hz under the ideal theorem. Practical systems normally include margin and anti-alias filtering.

Aliasing makes different continuous frequencies share sampled values

Sample a 900Hz sinusoid at1,000Hz. The discrete samples are indistinguishable in frequency from a100Hz alias under the usual baseband interpretation because900=1000−100.

Once aliasing has occurred in the sampled data, later digital filtering generally cannot reconstruct which out-of-band frequency generated it without additional information.

Record length and sampling rate set a Fourier frequency grid

For an N-point discrete Fourier transform of data sampled at fs, frequency-bin spacing is Δf=fs/N.

With fs=1024Hz and N=256, Δf=4Hz. With N=1024 at the same sample rate, spacing becomes1Hz.

Finer bin spacing does not automatically mean every pair of nearby physical tones can be resolved; window shape, record length, noise and estimator choice also matter.

Fourier analysis represents a signal through frequencies

A time-domain waveform may be complicated while its frequency representation is simple. For x(t)=3+2cos(2π10t), the constant term is a DC component of3 and the sinusoidal component has frequency10Hz and amplitude2.

MIT’s Signals and Systems material develops Fourier series, Fourier transforms and discrete-time Fourier representations as related ways of moving between time and frequency descriptions.

Spectral leakage comes from observing a finite record

If a sinusoid completes an integer number of cycles inside a rectangular observation window, its DFT energy can align neatly with a bin. If it does not, the finite cut creates energy across neighbouring bins.

Leakage is not random noise. It is a deterministic consequence of multiplying an infinite or longer signal by a finite observation window.

Windowing trades main-lobe width against sidelobes

Replacing a rectangular window with a tapered window can reduce distant spectral sidelobes while broadening the main lobe. There is no universal “best window” independent of the measurement goal.

The choice belongs to the analysis specification: detect a weak tone beside a strong one, estimate amplitude, locate frequency, or preserve time localisation.

Filtering multiplies frequency components by a response

If a linear filter has magnitude gain0.2 at a particular frequency and the input sinusoid amplitude is5, the output amplitude at that frequency is1 in the ideal steady-state model.

A filter can therefore attenuate some frequencies and preserve or amplify others. The full response includes both magnitude and phase.

Convolution is filtering in the time domain

For finite sequences x=[1,2,1] and h=[1,1], linear convolution is [1,3,3,1].

Each output sample is a weighted sum of shifted input samples. Fourier theory links this time-domain convolution to multiplication in the frequency domain.

A moving average is a simple finite impulse response filter

For data3,6,9,12,15, a three-point moving average gives6,9,12 when only complete windows are retained.

Smoothing reduces rapid variation but also blurs genuine fast changes. A smoother signal is not automatically a truer signal.

RMS measures quadratic signal size

For a zero-mean sinusoid with peak amplitude A, RMS=A/√2. A sinusoid with peak10 has RMS≈7.071.

RMS is useful because squaring prevents positive and negative portions from cancelling and connects naturally to power in many physical models.

Decibels convert ratios into logarithmic differences

For an amplitude ratio under compatible impedance assumptions, level difference can be 20log10(A2/A1). Doubling amplitude gives about6.02dB.

For a power ratio, the coefficient is10: a power ratio of100 gives20dB.

Signal-to-noise ratio requires a declared measure

If signal power is100 units and noise power1 unit, SNR=100 and SNRdB=10log10(100)=20dB.

Amplitude ratios and power ratios should not be mixed inside the same logarithmic formula without checking the physical relationship.

Averaging can reduce independent zero-mean noise

If n repeated measurements contain independent zero-mean noise with standard deviation σ, the mean has standard deviation σ/√n.

With σ=4 and n=16, standard deviation of the average is1. Systematic bias does not disappear under this rule.

Quantisation maps a continuous range to discrete codes

An ideal 8-bit converter has256 code levels. If a 0–5V range is divided uniformly, nominal step size is5/256≈0.01953V, or19.53mV, under one common endpoint convention.

More bits reduce ideal step size, but real converter accuracy also depends on noise, linearity, reference stability and front-end electronics.

Cascaded gains multiply while decibel gains add

Two stages with amplitude gains2 and0.5 have net gain1. In decibels they contribute approximately+6.02dB and−6.02dB, summing to0dB.

The logarithm converts multiplication into addition, which is why decibels are convenient for long signal chains.

Cross-correlation can estimate delay

If one sampled waveform is a shifted version of another, the lag that maximises cross-correlation can estimate the relative sample delay.

At fs=2,000Hz, a lag of6 samples corresponds to6/2000=0.003s or3ms.

Sampling duration sets the observation window

N=2048 samples at fs=1024Hz represent two seconds of data under the standard duration N/fs convention.

Longer observation can improve frequency-grid spacing but may violate stationarity if the signal changes during the record.

A complete signal-processing report preserves the acquisition chain

State sensor units, analogue bandwidth, sample rate, record length, quantisation, filtering, windowing and the metric used for amplitude or noise.

The return path is physical quantity → sensor → sampled sequence → mathematical transformation → interpreted feature. Every stage can add limits or uncertainty.

Practice: twenty signal-processing Mathematics questions

1. A sinusoid has f=5Hz. Find period.
2. fs=1000Hz. Find sampling interval.
3. How many samples are in2s at1000samples/s?
4. A signal is bandlimited to400Hz. What ideal minimum sampling boundary is implied by twice the bandwidth?
5. What baseband alias does900Hz produce when sampled at1000Hz?
6. For fs=1024Hz,N=256, find DFT bin spacing.
7. For fs=1024Hz,N=1024, find bin spacing.
8. Identify the DC value and sinusoid frequency in x=3+2cos(2π10t).
9. Convolve [1,2,1] with [1,1].
10. Find three-point moving averages of3,6,9,12,15.
11. A sinusoid has peak amplitude10. Find RMS.
12. Find amplitude-level change in dB for ratio2.
13. Find power-level change in dB for ratio100.
14. Signal power100, noise power1. Find SNR and SNRdB.
15. Independent noise SD4 is averaged over16 measurements. Find mean SD.
16. An ideal 8-bit range spans0–5V. Find nominal step5/256.
17. Amplitude gains2 and0.5 are cascaded. Find total gain.
18. At fs=2000Hz, convert lag6 samples to milliseconds.
19. N=2048 at fs=1024Hz. Find duration.
20. Why can a smoother filtered signal be less faithful to a fast real change?

Worked answers

1. 0.2s.
2. 0.001s=1ms.
3. 2,000.
4. 800Hz is the twice-bandwidth boundary; practical sampling should exceed it under the ideal theorem assumptions.
5. 100Hz.
6. 4Hz.
7. 1Hz.
8. DC=3; sinusoid frequency=10Hz.
9. [1,3,3,1].
10. 6,9,12.
11. 10/√2≈7.071.
12. About+6.02dB.
13. 20dB.
14. 100, or20dB.
15. 1.
16. About0.01953V=19.53mV.
17. 1.
18. 3ms.
19. 2s.
20. Because smoothing attenuates rapid variation whether that variation is noise or genuine signal structure.

Sources and connected applications

For continuous- and discrete-time signals, Fourier representations, filtering and sampling, see MIT OpenCourseWare: Signals and Systems and MIT 6.003 Signals and Systems. The numerical waveforms and measurements here are original teaching examples.

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