Continuous-time reconstruction

Summary

A signal can be reconstructed from time-discrete samples without error, given that the sampling rate is more than twice the highest frequency component appearing in the signal.

Introduction

A continuous-time (“analog”‘) waveform can be converted to a stream of discrete-time (“digital”) samples.
Given the sample stream, the analog waveform can be reconstructed.
This idea lies at the heart of digital signal processing.
But how can it be possible to recover the original waveform “in the gaps between samples”, ideally with perfect accuracy?

Bandwidth limit

Figure 1 shows the original analog signal (blue), that was sampled periodically (red):

Figure 1: Continuous-time signal (blue), discrete-time samples (red)

Knowing the samples alone is not enough to reconstruct the original waveform, as figure 2 shows: One could draw just about any signal through the given points.

Figure 2: A reconstruction attempt

Clearly, an additional constraint is required: The reconstructed signal should pass through the points “as smoothly as possible”.

This is achieved by limiting the bandwidth - in other words, forbid any signal energy beyond a cutoff frequency

Three cases are possible:

If is too high, there will still be many different curves that pass through the points
But choose a too small , and it will make it impossible to find any curve that connects the points
In-between, there is an optimum that allows only one single curve shape to pass through the points

is the well-known Nyquist rate:

Only a single curve through the samples is possible, as long as only frequencies are allowed that have more than two samples per periold.

.
With regard to the example above, having exactly two samples per period is not sufficient.

© Markus Nentwig 2007-2008
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Please send me a mail! mnentwig@elisanet.fi