by Sneha Das

ABSTRACT

In this article, we shall highlight upon the draw-backs of Fourier analysis, get a birds eye view of the Short Time Fourier Analysis and try understanding the Uncertainty Principle that governs signals.

As a prerequisite to be able to interpret this article better, the readers could read through some fundamentals of Fourier Analysis; This article might be helpful

https://eceforum.wordpress.com/2011/10/24/fourier-analysis-the-mathematical-prism/

INTRODUCTION

The Fourier analysis is one of the most boggling discoveries of the 19th century.

Even the most amazing revelations are not free from flaws, and so does the Fourier analysis. Fourier transforms do not cater any clear information regarding the time at which a particular frequency component is present in the signal. This poses as a serious gap in our understanding when the signal under consideration is of non-stationary in nature, or when the time details of the spectral components are required.

Thus, when the time localization of the spectral components is needed, a transform giving the Time-Frequency representation of the signal is necessary.

This is where Short Time Fourier Transforms (STFT) also known as Windowed Fourier Transform comes in.

In STFT, we assume that some portion of the non- stationary signal is stationary and periodic, and we perform Fourier transform on the stationary portion of the signal separately. By stationary, it is implied that it is a signal whose statistical properties do not change with time. That is the frequency content of the signal is constant at all times.

If the region where the signal can be assumed to be stationary is too small, then we look at that signal through narrow windows, narrow enough that the portion of the signal seen from these windows are indeed stationary.

Thus using STFT we get the time localization of the spectral analysis of a signal.

We would expect the problem to be solved.

Well, not yet!

Here comes the Uncertainty Principle of Signals.

This principle states that “One cannot know the exact time-frequency representation of a signal, i.e., one cannot know what spectral components exist at what instances of times. What one can know are the time intervals in which certain band of frequencies exist, which is a resolution problem.”

Sneak peek into the Uncertainty Principle of Signals

By resolution, what is being implied is that in the frequency domain we know exactly all the frequencies that exist. In the time domain the value of the signal at every point in time is well defined. That is because our window in Fourier Transform goes from – infinity to + infinity i.e. it exists at all times.

The problem with the STFT is the width of the window function. The window that we are using is of finite length. Thus it covers only a portion of the signal, which causes the frequency resolution to get poorer. What is implied by ‘getting poorer’ is that, we no longer know the exact frequency components that exist in the signal, but we only know a range of frequencies that exist.

You may as well question “why don’t we make the window width infinite?”

Then we loose all the time information that we acquired and end up with Fourier Transform rather than STFT.

Thus to sum up, we can say if we use a window of infinite length, we get Fourier Transform which gives a perfect frequency resolution but limited time resolution. On the other hand, if we focus on stationary property of the signal, we have to use a narrow window in accordance to the stationarity of the signal. Thus, narrower the window, the better time resolution and stationarity of the signal but the frequencies resolution gets poorer.

We shall not go into the mathematics of the Uncertainty Principle as yet, but we shall try and understand using examples.

Short Time Fourier Transform

A non-stationary signal

The figure above represents a non-stationary signal with four different frequencies at different points in time.

We will perform STFT on the above signal using windows of four different widths and see how it affects the time and frequency resolution of the signal.

The figure below represents windows of four different widths.

Windows of four different widths

STFT using Window 1

The figure above shows STFT using window 1. We observe that the time resolution of the signal is good with the peaks separated from each other in time. But in the frequency domain every peak covers a range of frequencies rather than indicating the exact frequencies contained in the signal during that time period.

STFT using the Window 2

Above is the representation of the STFT using window 2. We can see the peaks merging into each other in the time domain and in the frequency domain the frequency range is slightly condensed as compared to the previous window.

STFT using Window 3

Now making the window even wider we see in the time domain the resolution has become poorer. The peaks cannot be differentiated. But the frequency ranges have shrunken thus improving the frequency resolution greatly.

STFT using Window 4

Now for the last and the widest window, the frequency domain shows the exact frequency components contained in the signal but the time domain tells us nothing as to when these frequencies occur. Thus very poor time resolution.

Thus the above demonstration totally justifies as to how the uncertainty principle affects the time frequency representation of signals.

REFERENCES:

1)    www.wikipedia.org

2)    http://cnx.org

3)    http://www.numberwatch.co.uk

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