Analog vs Digital Signals

Introduction

While "continuous vs discrete" refers to the time axis, "analog vs digital" primarily concerns the amplitude axis — whether the signal values are continuous or restricted to specific levels. In everyday language, people often use "analog" and "digital" loosely, but in signal processing, the distinction is precise and important.

An analog signal has continuous values in both time and amplitude — like the voltage output of a microphone. A digital signal is discrete in both time and amplitude — like the data stored on a hard drive. Understanding the conversion between these two forms is fundamental to all modern electronics.

Analog Signals

An analog signal is continuous in both its independent variable (time) and its dependent variable (amplitude). It can take any value within a continuous range at any instant of time.

$$x(t) \in \mathbb{R}, \quad t \in \mathbb{R}$$

Characteristics of Analog Signals

• Infinite resolution — can represent infinitesimally small differences
• Susceptible to noise accumulation (each processing stage adds noise)
• Bandwidth-limited by physical components
• Cannot be stored perfectly in finite memory
• Directly represent physical quantities (voltage, current, pressure)

Examples

• Output of a thermocouple (temperature → voltage)
• Vinyl record groove (physical displacement represents audio)
• Analog radio transmission (AM/FM modulated carrier waves)
• Oscilloscope trace of a circuit voltage

Digital Signals

A digital signal is discrete in both time and amplitude. The amplitude values are restricted to a finite set of levels, typically represented as binary numbers.

$$x[n] \in \{L_0, L_1, L_2, \ldots, L_{2^b - 1}\} \quad \text{where } b = \text{bit depth}$$

Characteristics of Digital Signals

• Finite resolution determined by the number of bits (bit depth)
• Noise-immune during transmission (can distinguish 0 from 1)
• Can be stored, copied, and transmitted without degradation
• Require bandwidth for transmission (bits per second)
• Processed by digital hardware (CPUs, FPGAs, DSPs)

Examples

• CD audio (16-bit samples at 44.1 kHz)
• Digital temperature reading displayed as "23.5°C"
• JPEG image (8 bits per color channel per pixel)
• Digital communication (bits transmitted over fiber optic)

The Four Signal Categories

Combining time and amplitude classifications gives four categories:

1. Analog (CT, continuous amplitude): Natural physical signals
2. Sampled (DT, continuous amplitude): After ADC sampling, before quantization
3. Quantized (CT, discrete amplitude): Theoretical; rarely occurs alone
4. Digital (DT, discrete amplitude): What computers actually store and process

Analog-to-Digital Conversion (ADC)

Converting analog to digital involves three steps:

Step 1: Sampling

Capture the analog signal at regular intervals: $$x_s[n] = x(nT_s)$$

The sampling rate $f_s = 1/T_s$ must satisfy the Nyquist criterion: $f_s > 2f_{max}$.

Step 2: Quantization

Map each continuous sample value to the nearest allowed discrete level. With $b$ bits, there are $2^b$ quantization levels.

For a signal range $[-V_{max}, V_{max}]$: $$\Delta = \frac{2V_{max}}{2^b}$$

where $\Delta$ is the step size (resolution) between adjacent levels.

Quantization error: $e[n] = x_q[n] - x_s[n]$, bounded by $|e[n]| \leq \Delta/2$

The Signal-to-Quantization-Noise Ratio (SQNR): $$\text{SQNR} \approx 6.02b + 1.76 \text{ dB}$$

Each additional bit gives approximately 6 dB improvement. This is why 16-bit audio (96 dB dynamic range) sounds much better than 8-bit audio (48 dB).

Step 3: Encoding

Represent each quantized level as a binary number. The result is a stream of bits that can be stored, transmitted, or processed digitally.

Digital-to-Analog Conversion (DAC)

The reverse process reconstructs an analog signal from digital data:

1. Decode the binary values into amplitude levels
2. Hold each value for one sample period (zero-order hold)
3. Reconstruct by low-pass filtering to smooth the staircase output

The reconstruction filter removes the high-frequency "staircase" artifacts and produces a smooth analog signal.

Advantages and Disadvantages

Analog Signals

Advantages:

• Infinite resolution (no quantization noise)
• No conversion delay (real-time processing)
• Simple circuits for basic operations (amplification, filtering)

Disadvantages:

• Noise accumulates with each processing/copying stage
• Difficult to store long-term without degradation
• Hard to implement complex operations
• Component aging and temperature affect accuracy

Digital Signals

Advantages:

• Perfect storage and copying (bits don't degrade)
• Noise immunity (clear distinction between 0 and 1)
• Flexible processing (software-defined operations)
• Easy encryption and error correction
• Reproducible results regardless of hardware aging

Disadvantages:

• Finite resolution (quantization noise)
• Conversion introduces delay (latency)
• Requires higher bandwidth for transmission
• Complex hardware needed (processors, memory)

Practical Example: Audio Systems

Analog audio path (vinyl):

Each stage adds noise. The 50th copy of a tape sounds terrible.

Digital audio path (CD/streaming):

The digital portion introduces no additional noise. The millionth copy is identical to the first.

Bit Depth and Quality

Key Takeaways

• "Analog vs digital" is about amplitude resolution, not just time
• Digital signals sacrifice resolution for noise immunity and perfect reproduction
• ADC involves sampling → quantization → encoding
• Each bit of resolution adds ~6 dB of dynamic range
• Modern systems overwhelmingly prefer digital processing with analog interfaces (ADC/DAC)
• The quality tradeoff is: more bits = better quality but more data to store/transmit
• Understanding both domains is crucial because real systems always bridge them