Mathematics - Signals And Systems - Interpolation


[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, xp(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 ωc, 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

  1. 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

LTI Systems and Convolution

Fourier Series and Transform

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!

H2
H3
H4
3 columns
2 columns
1 column
1 Comment