Mathematics - Signals And Systems - Interpolation

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

Alan Oppenheim. RES.6-007 Signals and Systems. Spring 2011. Massachusetts Institute of Technology: MIT OpenCourseWare, License: Creative Commons BY-NC-SA.

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