The delay introduced by a digital filter

The rounding error in each sample

The loss of amplitude when a signal is digitised

E1 · Published 26 September 2026 · By The Ham Desk

Digital Signal Processing: Sampling, Quantising and Filtering

A modern transceiver converts the radio signal to numbers, does everything to the numbers, and converts them back. This unit covers the rules that conversion must obey, and what happens when it does not.

Band none
Study time 18
Level Core
questions 10

Waveform and antenna diagram for Signal Processing and Digital Communications — Signal Processing and Digital Communications — waveform overlay and radiation pattern

Pre-flight briefing

Two rules govern the conversion

The sampling theorem says that a signal must be sampled at least twice as fast as the highest frequency it contains. Sample at 8,000 samples per second and the highest frequency that can be represented unambiguously is 4,000 Hz - the Nyquist frequency. Anything above that does not disappear; it folds back into the band and appears at a frequency that is not really there, an effect called aliasing. A 5 kHz component sampled at 8 kHz reappears as a 3 kHz tone, and once it has been aliased there is no way to tell it apart from a genuine 3 kHz signal.

That is why a low-pass filter precedes the analogue-to-digital converter. The anti-aliasing filter removes energy above the Nyquist frequency before conversion, because after conversion it is too late. On the output side a reconstruction filter does the same job in reverse, smoothing the staircase of samples back into a continuous waveform.

The second rule concerns amplitude. Quantisation assigns each sample to the nearest of a finite set of levels, and the error is at most half a step. Each additional bit doubles the number of levels and improves the signal-to-noise ratio by about 6 dB, so an ideal 16-bit converter has roughly 96 dB of dynamic range while an 8-bit converter has about 48 dB - which is why a receiver's performance is often described by its ADC resolution.

What the numbers then allow

Once the signal is a sequence of numbers, filtering becomes arithmetic. A digital filter is a set of multiplications and additions whose coefficients determine the response: it can be linear phase, which no analogue filter achieves exactly, it can have a bandwidth of a few hertz at audio rates, and it never drifts with temperature or age because its coefficients are stored values. Changing the coefficients changes the filter - which is how one piece of hardware becomes a dozen different bandwidths in a modern receiver.

The fast Fourier transform converts a block of samples from the time domain to the frequency domain, computing what a spectrum analyser displays. In a receiver it produces the waterfall and panadapter display; in a transmitter it is used for modulation and for shaping. Because it processes a block of samples at once, it is a batch calculation rather than a continuous process, and its resolution depends on how many samples are in the block.

Decimation reduces the sample rate by discarding samples in a controlled way after filtering, which eases the processing load once a narrow signal has been isolated. Interpolation does the reverse, raising the effective rate before conversion back to analogue. Together they allow a receiver to spend its computation where the signal actually is.

Why this replaced the analogue chain

A software defined radio moves the boundary between hardware and software as far toward the antenna as the technology allows: instead of mixing, filtering and detecting with discrete components, it digitises a wide slice of spectrum and does all of it arithmetically. The advantages are repeatability, flexibility and the ability to implement filters and detectors that have no practical analogue equivalent.

The costs are real too. A wideband front end accepts every strong signal in the band it digitises, so it needs a high dynamic range and good linearity; a converter that clips on a strong nearby signal generates spurious responses across the whole captured spectrum. Digital processing also consumes power and generates its own clock-related spurs, which is why the analogue front end has not disappeared - it has simply moved.

Terms you must be able to define
Term	What the examiner wants to hear
Nyquist frequency	Half the sampling rate: the highest frequency that can be represented without aliasing.
Aliasing	Folding of signals above the Nyquist frequency back into the band as false responses.
Quantisation	Assigning each sample to one of a finite set of levels; about 6 dB of dynamic range per bit.
Anti-aliasing filter	A low-pass filter placed before the ADC to remove energy above the Nyquist frequency.
FFT	Fast Fourier transform: converts a block of samples from the time to the frequency domain.

Question deck 10 questions

E2A01 Recall

What does the sampling theorem require?

E2A02 Core

What is aliasing?

E2A03 Core

What is the purpose of an anti-aliasing filter in a digital receiver?

E2A04 Core

How much dynamic range does each additional bit of an analogue-to-digital converter add?

E2A05 Core

What does a fast Fourier transform produce from a block of samples?

E2A06 Core

What is a significant advantage of a digital filter over an analogue one?

E2A07 Stretch

What is decimation in a digital signal processing chain?

E2A08 Stretch

What is a principal risk of a wideband software defined radio front end?

E2A09 Core

What is quantisation error and how is it reduced?

E2A10 Stretch

Why does a modern receiver often describe its performance in terms of ADC resolution?

Reference sheet

The explanations for all ten questions sit on this sheet. Pick an answer on the left and the matching card lights up.

E2A01 Explanation 1

Key D — A signal must be sampled at least twice as fast as the highest frequency it contains

Sampling at exactly twice the highest frequency is the minimum; anything less and the high components fold back as aliases. In practice converters sample well above the minimum, both to leave room for the anti-aliasing filter's transition band and to simplify that filter's design.

Rule Fundamentals - sampling theory

E2A02 Explanation 2

Key C — Signals above the Nyquist frequency appearing at false lower frequencies after sampling

Aliasing is a one-way door: once a component has folded down into the band it is indistinguishable from a real signal there, so no amount of later processing recovers it. The only defence is filtering before the converter.

Rule Fundamentals - aliasing

E2A03 Explanation 3

Key B — It attenuates signals above the Nyquist frequency before they reach the analogue-to-digital converter

Filtering must happen in the analogue domain and before conversion, because after conversion the alias and a genuine signal are arithmetically identical. This is why a receiver's front-end filter and its converter sample rate have to be designed together.

Rule Fundamentals - anti-aliasing filters

E2A04 Explanation 4

Key A — About 6 dB

Doubling the number of levels halves the quantisation step and improves the signal-to-quantisation-noise ratio by about 6 dB, which is why 16-bit audio has roughly 96 dB of range and 24-bit roughly 144 dB. Practical converters fall short of the ideal because of their own noise and distortion.

Rule Fundamentals - quantisation

E2A05 Explanation 5

Key D — A representation of the signal's frequency content over that block

The FFT maps a block of time samples to a spectrum, which is what fills a panadapter or waterfall display. Because it works on a finite block, its frequency resolution is set by the block length and its output is only valid for the time interval the block covers.

Rule Fundamentals - DSP

E2A06 Explanation 6

Key C — Its response is set by stored coefficients, so it does not drift and can be changed at will

Because the filter is arithmetic, identical hardware can implement any response its coefficients describe, and the response is the same on the day of manufacture and ten years later. That repeatability is why DSP replaced so much of the analogue signal chain.

Rule Fundamentals - digital filters

E2A07 Explanation 7

Key B — Reducing the sample rate by discarding samples after filtering

After a narrow signal has been isolated, most of the remaining samples carry no new information, so decimating reduces the processing load without losing the signal. Decimation must follow filtering, because discarding samples first would alias the wideband content down into the band of interest.

Rule Fundamentals - sample rate conversion

E2A08 Explanation 8

Key A — A strong nearby signal can drive the converter into clipping and generate spurious responses across the captured spectrum

Once a converter clips, the resulting intermodulation and harmonic products appear at frequencies that have nothing to do with the interfering signal, so the whole captured band is polluted. This is why wideband receivers need high dynamic range converters and well-designed front-end filtering.

Rule Fundamentals - receiver dynamic range

E2A09 Explanation 9

Key D — The difference between the actual sample value and the nearest representable level; it is reduced by using more bits

Quantisation error is at most half a step, and the step halves with every extra bit - which is the same statement as 6 dB of dynamic range per bit. It sets the noise floor below which a digital receiver cannot hear, whatever the analogue front end does.

Rule Fundamentals - quantisation noise

E2A10 Explanation 10

Key D — The converter's dynamic range sets the weakest signal the receiver can copy in the presence of strong ones

In a software defined radio, everything downstream of the converter is arithmetic, so the converter's dynamic range and linearity become the practical limit on how well the receiver performs in a crowded band. That is why the specification is quoted so prominently.

Rule Fundamentals - software defined radio

How to use this unit

Work the ten questions in one sitting with the reference sheet covered, then uncover it and read every explanation - including the ones you got right. The wording of the real exam is different, so learn the reasoning rather than the answer letters.