Time, Equal, Volts, Temperature, Signal: Digital Signals Explained

Introduction

In this blog, I will walk through the fundamentals of digital signals – what they are, how they differ from discrete-time signals, and why we need them. We’ll look at real examples involving temperature and voltage, and I’ll show you how increasing quantization levels reduces error.

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Why Digital Signals? The Real Question

Before diving into definitions, let’s address the question that often comes last: if analog and discrete-time signals already exist, why do we need digital signals at all? I’ll answer this directly.

Digital signals are essential because they allow robust storage, processing, and transmission without degradation. In an analog system, noise and component tolerances accumulate. In a digital system, once we discretize magnitude into fixed levels, we can regenerate the exact levels at each step. This is why your music on a CD sounds the same every time, while a cassette tape wears out. I will explore this more in the next post, but for now, know that the need arises from noise immunity and reproducibility.

Discretizing Both Axes: Time and Magnitude

Let me define the core concept. In discrete-time signals, we only sample time at equal intervals. The magnitude (amplitude) can still take any continuous value. In digital signals, we also force the magnitude to take only specific, allowed levels.

Equal Time Intervals

We start by dividing the time axis into equal segments. If we label time points as \( t_0, t_1, t_2, \dots, t_N \), then the time interval \( \Delta t \) is constant: \[ \Delta t = t_1 - t_0 = t_2 - t_1 = \dots = t_N - t_{N-1} \]

This is no different from discrete-time sampling. The new step is the quantization of the magnitude axis.

Quantization – The Magnitude Axis

I will illustrate with a new example, not the one from the video. Suppose I am measuring the temperature inside an industrial kiln over time. The temperature ranges from 0°C to 100°C. For a digital signal, I must decide on allowed levels. Let’s say I start with only 4 levels: 0°C, 25°C, 50°C, 75°C, and 100°C.

Now I measure actual temperatures at five time instants:

• At \( t_1 \), actual = 22°C. Allowed levels: 0 or 25. The lower level is 0? Wait, careful. The rule I explained earlier: to minimize error, we select the lower level (floor quantization). 22°C is between 0 and 25 – the lower is 0°C. So digital value = 0°C.
• At \( t_2 \), actual = 48°C. Between 25 and 50? Actually 48 is between 25 and 50? No, between 25 and 50? 48 > 25, <50, lower is 25°C. Digital = 25°C.
• At \( t_3 \), actual = 74°C. Between 50 and 75? Lower = 50°C.
• At \( t_4 \), actual = 100°C. Exactly on a level → 100°C.
• At \( t_5 \), actual = 1°C. Between 0 and 25 → lower = 0°C.

This shows the key property: the signal can only take values equal to those preset levels. The error is the difference between actual and the quantized value.

The Voltage Example – Error Reduction by Increasing Levels

Let me take a different scenario: measuring a signal voltage that varies between 0V and 5V. Initially, suppose I only allow two levels: 0V and 5V. At time \( t_a \), the actual voltage is 2.2V. The lower level is 0V. Error = 2.2V. That is huge.

Now I increase the number of levels. I divide the 0–5V range into 5 equal parts: each step = 1V. Allowed levels: 0, 1, 2, 3, 4, 5V. The same 2.2V actual voltage falls between 2V and 3V. Lower level = 2V. Error = 0.2V. Much smaller.

If I increase further – say 0, 0.5, 1.0, 1.5, … up to 5V (11 levels) – then 2.2V lies between 2.0V and 2.5V. Lower level = 2.0V still, error 0.2V. To get exact 2.2V, I would need an infinite number of levels, but in practice we choose enough levels to make the error acceptable (e.g., 8-bit quantization gives 256 levels).

Key takeaway: As the number of levels increases, the quantization error decreases. In the limit (infinite levels), we return to a continuous-amplitude signal, but that defeats the purpose of digital. We choose a finite number based on required fidelity.

Discrete-Time vs Digital – A Side-by-Side

Let me summarize the distinction using an audio recording example (a different context).

• Discrete-time signal: I sample my voice every 0.1 milliseconds, but each sample can have any amplitude value (e.g., 0.1234567 volts). Storing this requires infinite precision.
• Digital signal: I sample every 0.1 milliseconds and I round each sample to the nearest allowed voltage level, say one of 65536 levels (16-bit). Now each sample is represented by a fixed number of bits.

The table below contrasts them:

Feature	Discrete-time signal	Digital signal
Time axis	Discrete (equal intervals)	Discrete (equal intervals)
Magnitude axis	Continuous (any value)	Discrete (specific levels)
Storage	Requires infinite bits	Finite bits (e.g., 8, 16)
Noise immunity	Low (noise changes value)	High (noise must push value past a level boundary)

Putting It All Together – A Practical Rule

When you design a digital signal processing system, you must choose:

1. Sampling interval \( \Delta t \) (determines Nyquist frequency)
2. Number of quantization levels \( L \) (determines dynamic range and error)

The quantization error for a signal uniformly distributed between levels is at most half the step size. If the step size is \( q \), the maximum error is \( q \). By taking the lower level (floor), my error is always between 0 and \( q \). Some systems use rounding to nearest (which gives symmetric error), but the principle is the same: more levels → smaller \( q \) → smaller error.

Final Thought – And a Question for You

We’ve covered that digital signals require equal time intervals and fixed magnitude levels. The lower-level rule minimizes error in the floor-quantization scheme I used. Increasing levels improves accuracy.

Now, I asked earlier: why digital? You might think analog can be more accurate (infinite levels), but digital offers repeatability, error correction, and easy processing. In my next post, I’ll give concrete examples of systems that switched from analog to digital and why they succeeded.

I want to hear from you: Can you think of a real-world device that still uses pure analog signals? And why hasn’t it gone digital? Post your answer in the comments below.