[Image1]

## Introduction

Hey it's a me again @drifter1!

Today we continue with my mathematics series about **Signals and Systems** in order to cover **Interpolation**.

So, without further ado, let's get straight into it!

## Reconstruction Procedure

In the previous article we based the reconstruction of continuous-time signals that have been sampled on the use of an ideal low-pass filter.
This is a natural consequence of interpreting sampling as a procedure in the frequency domain.
To get more specific, reconstruction in the time domain is basically a convolution of the impulse train of samples, *x _{p}(t)*, with the impulse response of the low-pass filter,

*h(t)*:

## Interpolation

The convolution of the samples with the impulse response of the low-pass filter can also be viewed as a superposition of weighted delayed impulse responses, with amplitudes and positions that correspond to the impulses in the impulse train. This superposition represents an interpolation procedure between the samples.

### Band-Limited

When the reconstruction filter is ideal, the interpolation function is a sinc function, and the process is referred to as band-limited interpolation.
For such an ideal low-pass filter, with cutoff frqeuency *ω _{c}*, the impulse response (or interpolation function) is:

### Zero-Order Hold

In addition to band-limited interpolation, there are also other commonly used interpolation procedures. For example, there is zero-order hold, which interpolates between sample points by holding each sample value until the next sampling instant. This leads to a staircase-like approximation of the original signal. Zero-order hold corresponds to a convolution with a rectangular pulse interpolation function, with a pulse duration that is equal to the sampling period.

[Image 2]

### First-Order Hold (or Linear Interpolation)

Next up there is also linear interpolation, which is also referred to as first-order hold. In this type of interpolation the sample points are interconnected by straight line segments. It basically corresponds to a triangle function with a duration that is exactly twice the sampling period.

[Image 3]

## RESOURCES:

### References

### Images

Mathematical equations used in this article were made using quicklatex.

Block diagrams and other visualizations were made using draw.io and GeoGebra

## Previous articles of the series

### Basics

- Introduction → Signals, Systems
- Signal Basics → Signal Categorization, Basic Signal Types
- Signal Operations with Examples → Amplitude and Time Operations, Examples
- System Classification with Examples → System Classifications and Properties, Examples
- Sinusoidal and Complex Exponential Signals → Sinusoidal and Exponential Signals in Continuous and Discrete Time

### LTI Systems and Convolution

- LTI System Response and Convolution → Linear System Interconnection (Cascade, Parallel, Feedback), Delayed Impulses, Convolution Sum and Integral
- LTI Convolution Properties → Commutative, Associative and Distributive Properties of LTI Convolution
- System Representation in Discrete-Time using Difference Equations → Linear Constant-Coefficient Difference Equations, Block Diagram Representation (Direct Form I and II)
- System Representation in Continuous-Time using Differential Equations → Linear Constant-Coefficient Differential Equations, Block Diagram Representation (Direct Form I and II)
- Exercises on LTI System Properties → Superposition, Impulse Response and System Classification Examples
- Exercise on Convolution → Discrete-Time Convolution Example with the help of visualizations
- Exercises on System Representation using Difference Equations → Simple Block Diagram to LCCDE Example, Direct Form I, II and LCCDE Example
- Exercises on System Representation using Differential Equations → Equation to Block Diagram Example, Direct Form I to Equation Example

### Fourier Series and Transform

- Continuous-Time Periodic Signals & Fourier Series → Input Decomposition, Fourier Series, Analysis and Synthesis
- Continuous-Time Aperiodic Signals & Fourier Transform → Aperiodic Signals, Envelope Representation, Fourier and Inverse Fourier Transforms, Fourier Transform for Periodic Signals
- Continuous-Time Fourier Transform Properties → Linearity, Time-Shifting (Translation), Conjugate Symmetry, Time and Frequency Scaling, Duality, Differentiation and Integration, Parseval's Relation, Convolution and Multiplication Properties
- Discrete-Time Fourier Series & Transform → Getting into Discrete-Time, Fourier Series and Transform, Synthesis and Analysis Equations
- Discrete-Time Fourier Transform Properties → Differences with Continuous-Time, Periodicity, Linearity, Time and Frequency Shifting, Conjugate Summetry, Differencing and Accumulation, Time Reversal and Expansion, Differentation in Frequency, Convolution and Multiplication, Dualities
- Exercises on Continuous-Time Fourier Series → Fourier Series Coefficients Calculation from Signal Equation, Signal Graph
- Exercises on Continuous-Time Fourier Transform → Fourier Transform from Signal Graph and Equation, Output of LTI System
- Exercises on Discrete-Time Fourier Series and Transform → Fourier Series Coefficient, Fourier Transform Calculation and LTI System Output

### Filtering, Sampling, Modulation, Interpolation

- Filtering → Convolution Property, Ideal Filters, Series R-C Circuit and Moving Average Filter Approximations
- Continuous-Time Modulation → Getting into Modulation, AM and FM, Demodulation
- Discrete-Time Modulation → Applications, Carriers, Modulation/Demodulation, Time-Division Multiplexing
- Sampling → Sampling Theorem, Sampling, Reconstruction and Aliasing

## Final words | Next up

And this is actually it for today's post!

Next time we will cover how continuous-time signals are processed as discrete-time signals...

See Ya!

Keep on drifting!