Digital Sound Representation and Sampling

Defining Digital Quality: Sampling Rate and Bit Depth

The quality and accuracy of a digital sound file depend on two main factors: how often we sample the sound and how precisely we measure each sample.

Sampling Rate

The sampling rate is the number of sound samples taken per second. It is the frequency at which the ADC "looks" at the analogue wave. The sampling rate is measured in Hertz (Hz), where $1\text{ Hz}$ is equal to one sample per second. A higher sampling rate means the digital representation is closer to the original continuous wave.

Sampling Resolution (Bit Depth)

While the sampling rate tells us how often we measure, the sampling resolution (often called bit depth) tells us how accurately we measure. Bit depth is the number of bits used to store each individual sample.

Exam Tip: Increasing either the sampling rate or the bit depth will increase the accuracy and quality of the sound, but it will also significantly increase the resulting file size.

Calculating File Size

Since digital sound is just a collection of binary numbers, we can calculate exactly how much storage space a sound file will occupy. To do this, we need to know the sampling rate, the bit depth (resolution), and the duration of the recording.

The formula for calculating the file size in bits is: $\text{File Size (bits)} = \text{Sampling Rate} \cdot \text{Sample Resolution} \cdot \text{Duration (in seconds)}$

Worked Examples

Example 1: Basic Calculation Calculate the file size in bits for a 30-second sound recording that has a sampling rate of $1000\text{ Hz}$ and a sample resolution of $4\text{ bits}$.

$\text{File Size} = 1000 \cdot 4 \cdot 30$ $\text{File Size} = 120,000\text{ bits}$

Example 2: Handling Minutes Calculate the file size in bits for a 2-minute sound recording that uses a sampling rate of $200\text{ Hz}$ and a sample resolution of $8\text{ bits}$. Note: First, convert minutes to seconds ($2 \cdot 60 = 120\text{ seconds}$).

$\text{File Size} = 200 \cdot 8 \cdot 120$ $\text{File Size} = 192,000\text{ bits}$

Example 3: Converting to Bytes Calculate the file size in bits for a 3-minute sound recording that uses a sampling rate of $2000\text{ Hz}$ and a sample resolution of $4\text{ bits}$. Convert your answer to bytes. Step 1: Convert minutes to seconds ($3 \cdot 60 = 180\text{ seconds}$). Step 2: Calculate bits. $\text{File Size (bits)} = 2000 \cdot 4 \cdot 180$ $\text{File Size (bits)} = 1,440,000\text{ bits}$ Step 3: Convert bits to bytes (divide by 8). $\text{File Size (bytes)} = \frac{1,440,000}{8}$ $\text{File Size (bytes)} = 180,000\text{ bytes}$