Scaling and linearity

Understanding Scaling and Linearity in Digital

Signal Processing (DSP)

Digital Signal Processing, or DSP, is the method we use to work with digital signals — like audio, images, and sensor data — using computers or digital devices. When working with signals, we often talk about how systems (like filters or amplifiers) change these signals.

Two important ideas in DSP are scaling and linearity. These are simple but powerful concepts that help us understand and design systems that handle signals in a predictable way.

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 What is a System in DSP?

Before we talk about scaling and linearity, let's understand what a system is.
In DSP, a system is anything that takes an input signal and gives an output signal. For example, a microphone is a system that takes in sound and gives an electrical signal.

We use notation like this:

$$x[n] \rightarrow y[n]$$

Where:

• $x[n]$ is the input signal (like your voice)

• $y[n]$ is the output signal (like the digital version of your voice)

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 What is Linearity?

A system is linear if it follows two simple rules:

1. Additivity

If the system processes two signals separately, then it should give the same result if we add the two signals first and then process them.

Example:

If:

• $x_1[n] \rightarrow y_1[n]$

• $x_2[n] \rightarrow y_2[n]$

Then:

• $x_1[n] + x_2[n] \rightarrow y_1[n] + y_2[n]$

In simple terms:
The system treats added signals the same as adding the results after processing.

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2. Homogeneity (or Scaling)

If you multiply the input signal by a number, the output should also get multiplied by the same number.

Example:

If:

• $x[n] \rightarrow y[n]$

Then:

• $3x[n] \rightarrow 3y[n]$

This means the system doesn’t change the shape of the signal, just the size (amplitude).

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 If a system satisfies both rules, it is called a linear system.

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 What is Scaling?

Scaling in DSP usually means changing the size (amplitude) of the signal.

Let’s say:

• $x[n] = [1, 2, 3]$

• Then $2x[n] = [2, 4, 6]$

We are simply multiplying every value of the signal by 2. This increases the volume of a sound signal or brightness in an image.

Sometimes, we also talk about time scaling, which means speeding up or slowing down the signal in time. But this is more common in continuous (analog) signals. In digital (discrete) signals, time scaling is harder because time is counted in whole steps.

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 Simple Example to Check Linearity

Let’s test a system to see if it’s linear.

System:

$$y[n] = 2x[n]$$

Test Additivity:

• Input 1: $x_1[n] = [1, 2]$, Output: $y_1[n] = [2, 4]$

• Input 2: $x_2[n] = [3, 4]$, Output: $y_2[n] = [6, 8]$

• Add inputs: $x_1[n] + x_2[n] = [4, 6]$

• Output: $y[n] = 2 \cdot [4, 6] = [8, 12]$

• Add outputs: $y_1[n] + y_2[n] = [8, 12]$

 Same result — Additivity passed

Test Scaling:

• Input: $x[n] = [1, 2]$

• Output: $y[n] = [2, 4]$

• Scale input: $3x[n] = [3, 6]$

• Output: $y[n] = 2 \cdot [3, 6] = [6, 12]$

 Scaling passed

Conclusion: This system is linear.

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 Example of a Non-Linear System

Try this system:

$$y[n] = x[n] + 5$$

If:

• Input: $x[n] = [1, 2]$

• Output: $y[n] = [6, 7]$

Now scale input:

• $2x[n] = [2, 4]$

• Output: $y[n] = [7, 9]$ — NOT double of original output

 Scaling failed
So, this system is not linear.

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 Why Is Linearity Important?

Linearity helps engineers and scientists:

• Predict system behavior easily

• Break complex signals into simple parts (like sine waves)

• Use tools like the Fourier Transform for frequency analysis

• Design filters that clean or enhance signals

In simple words:
Linear systems are easier to work with and understand.

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 Summary

In digital signal processing (DSP), scaling and linearity are key concepts that help us understand how systems handle signals. Scaling means changing the amplitude of a signal by multiplying it with a constant, making the signal larger or smaller. Linearity refers to a system’s ability to follow two rules: additivity, where the response to a sum of signals is the sum of the responses, and homogeneity, where scaling the input also scales the output by the same amount. If both of these conditions are met, the system is called linear. Linear systems are useful because they behave in predictable ways, making it easier to analyze and design signal processing tasks. These concepts are used in many real-world applications like filtering noise from audio, adjusting image brightness, or analyzing sensor data. Understanding scaling and linearity is essential for working effectively with digital signals.