The diagram shows an analogue sound wave. The Nyquist theorem states that the sample rate must be at least twice the highest frequency in the sound being re...

(a) The Nyquist theorem requires the sample rate to be at least twice the highest frequency in the sound because the digital recording must capture enough data points to accurately reconstruct the shape of the original analogue wave. If the sample rate falls below this threshold, the rapid oscillations of high-frequency sounds are not captured often enough. [1] This leads to a phenomenon called aliasing, where high-frequency components are misrepresented as lower frequencies in the digital recording, producing audible distortion that was not present in the original sound. [1]

(b) Since human hearing extends up to approximately 20,000 Hz, the minimum sample rate must be:

2 x 20,000 = 40,000 Hz [1]

(c) CD-quality audio uses 44,100 Hz rather than exactly 40,000 Hz because the additional margin above the theoretical minimum provides a buffer zone for anti-aliasing filters. These filters remove frequencies near the upper boundary that could otherwise cause aliasing artefacts, and they need a transition band to work effectively. [1]

(d) The calculation proceeds as follows:

Duration in seconds: 5 x 60 = 300 seconds

File size in bits: 8,000 x 8 x 300 = 19,200,000 bits

Convert to bytes: 19,200,000 / 8 = 2,400,000 bytes

Convert to kilobytes: 2,400,000 / 1,000 = 2,400 KB [1] [1]

(e) Telephone audio quality is noticeably lower than CD quality for two reasons. First, the telephone sample rate of 8,000 Hz can only capture frequencies up to 4,000 Hz (by the Nyquist theorem), which cuts off much of the higher-frequency detail in human speech, making it sound muffled compared to CD audio that captures up to 22,050 Hz. [1] Second, the 8-bit depth provides only \(2^8 = 256\) possible amplitude levels per sample, compared to \(2^{16} = 65,536\) levels for CD audio at 16-bit depth. This coarser quantisation introduces more quantisation noise and produces a less smooth, less natural reproduction of the sound. [1]