Sampling Rate and Quality in Digital Conversion Revision Notes | A Level AQA Physics

1. Why Digitising Analogue Signals Matters

Every time you stream music, make a phone call, or record a video, an analogue signal — a continuously varying wave — is being converted into a stream of numbers that a computer can store and transmit. The question is: how faithfully can we capture that wave in digital form?

The answer depends on two decisions made at the moment of conversion: how often you measure the signal, and how precisely you record each measurement. These two choices sit at the heart of this topic, and understanding them will let you explain, calculate, and evaluate digital audio and signal quality with confidence.

2. Sampling Rate — How Often You Listen

Picture a smoothly curving sound wave on an oscilloscope screen. To convert it to digital, you take a snapshot of its amplitude at regular time intervals. Each snapshot is called a sample, and the number of snapshots taken every second is the sampling rate, measured in hertz (Hz).

Think of it like a flip-book animation. If you draw only a few frames per second, the motion looks jerky and unnatural. Draw many frames per second and the motion appears smooth and continuous. A higher sampling rate gives you more data points to reconstruct the original wave, so the digital version more closely matches the analogue original.

There is a critical lower limit on sampling rate. The Nyquist criterion states that the sampling rate must be at least twice the highest frequency present in the signal. Human hearing extends to roughly 20 000 Hz, which is why CD-quality audio uses a sampling rate of 44 100 Hz — comfortably above the 40 000 Hz minimum required. If the sampling rate falls below twice the signal frequency, a distortion called aliasing occurs, where the reconstructed signal contains false lower-frequency components that were never in the original.

3. Bit Depth — How Precisely You Record Each Sample

Knowing how often to sample is only half the story. Once you have taken a sample, you need to record its amplitude as a binary number. The number of bits used to represent each sample is called the bit depth (or number of bits per sample).

With $n$ bits per sample, you can represent $2^n$ distinct amplitude levels. With 2 bits you have only 4 levels — a very coarse staircase approximation of the wave. With 8 bits you have 256 levels, and with 16 bits you have 65 536 levels. The more levels available, the more precisely each sample amplitude can be recorded, and the smaller the quantisation error — the difference between the true analogue amplitude and the nearest available digital level.

Quantisation error manifests as a background hiss or graininess in the reconstructed signal. Doubling the bit depth does not halve the error — each additional bit halves the size of each quantisation step, so the error reduces by a factor of two for every extra bit. This is why moving from 8-bit to 16-bit audio produces a dramatic improvement in perceived quality.

4. Pulse Code Modulation — Putting It All Together

Now that sampling rate and bit depth are clear, we can describe the complete process by which an analogue signal becomes a digital data stream. That process is called pulse code modulation, or PCM.

Here is the process in sequence. First, the analogue signal enters an analogue-to-digital converter (ADC). At each sampling instant, the ADC measures the instantaneous amplitude of the signal. That amplitude is then compared against the available quantisation levels — the number of which depends on the bit depth — and the nearest level is selected. Finally, that level is encoded as a binary number with $n$ bits, producing one pulse code. Repeat this at every sampling instant and you have a continuous stream of binary pulses: the PCM signal.

• Step 1 — Sample: measure the analogue amplitude at each time interval set by the sampling rate.
• Step 2 — Quantise: round the measured amplitude to the nearest available level (one of $2^n$ levels).
• Step 3 — Encode: convert the quantisation level to an $n$-bit binary number.
• Step 4 — Transmit or store: send or record the binary stream.

At the receiving end, a digital-to-analogue converter (DAC) reads each binary code, reconstructs the corresponding amplitude level, and outputs a stepped waveform. A low-pass filter then smooths the steps into a continuous analogue signal. The closer the sampling rate and bit depth are to the original signal's requirements, the more faithful this reconstruction will be.

5. Bit Rate and File Size — The Cost of Quality

Higher sampling rates and greater bit depths both improve signal quality, but they come at a cost: more data. The rate at which data is generated is called the bit rate, and it determines both the bandwidth needed to transmit the signal and the storage space needed to record it.

For a stereo signal (two channels), multiply the bit rate by 2. To find the file size for a recording of duration $t$ seconds, multiply the bit rate by $t$ to get the total number of bits, then divide by 8 to convert to bytes.

Worked example: Calculating Bit Rate and File Size for a PCM Audio Recording

A mono audio signal is digitised using PCM with a sampling rate of 44 100 Hz and a bit depth of 16 bits per sample. Calculate (a) the bit rate and (b) the file size in megabytes (MB) for a 3-minute recording. (1 MB = 1 000 000 bytes)

1. $$\text{Bit rate} = f_s \times n$$ — Bit rate equals sampling rate multiplied by bit depth. [M1]
2. $$\text{Bit rate} = 44\,100 \times 16$$ — $f_s = 44\,100$ Hz, $n = 16$ bits per sample. [M1]
3. $$44\,100 \times 16 = 705\,600 \text{ bps}$$ — Working: $44\,100 \times 10 = 441\,000$; $44\,100 \times 6 = 264\,600$; $441\,000 + 264\,600 = 705\,600$ bps. [A1]
4. $$t = 3 \times 60 = 180 \text{ s}$$ — 3 minutes multiplied by 60 seconds per minute.
5. $$\text{Total bits} = 705\,600 \times 180 = 127\,008\,000 \text{ bits}$$ — Working: $705\,600 \times 180 = 705\,600 \times 100 + 705\,600 \times 80 = 70\,560\,000 + 56\,448\,000 = 127\,008\,000$ bits.
6. $$\text{Total bytes} = \frac{127\,008\,000}{8} = 15\,876\,000 \text{ bytes}$$ — Divide by 8 because there are 8 bits in one byte. [M1]
7. $$\text{File size} = \frac{15\,876\,000}{1\,000\,000} = 15.876 \text{ MB}$$ — Divide by 1 000 000 as stated in the question. [A1] Sanity check: CD-quality audio runs at roughly 10 MB per minute, so ~16 MB for 3 minutes is consistent.

Answer: (a) Bit rate = 705 600 bps (approximately 706 kbps). (b) File size ≈ 15.9 MB.

6. Comparing the Effects of Sampling Rate and Bit Depth on Quality

Both parameters affect quality, but they do so in different ways. It is worth being precise about this distinction, because exam questions frequently ask you to compare or explain the effect of each one separately.

A low sampling rate causes the digital signal to miss rapid changes in the analogue waveform. High-frequency components of the signal are lost or distorted, making the reconstructed signal sound muffled or producing aliasing artefacts. Increasing the sampling rate captures faster variations more accurately, preserving higher-frequency content.

A low bit depth causes large quantisation errors because the amplitude of each sample must be rounded to one of only a few coarse levels. This introduces a background noise — quantisation noise — that is most noticeable during quiet passages. Increasing the bit depth reduces the size of each quantisation step, lowering quantisation noise and improving dynamic range.

7. Exam Technique and Common Mistakes

This topic rewards precise language. Examiners distinguish between students who understand the mechanism and those who have memorised vague phrases. The guidance below targets the specific errors that cost marks most often.