The Fourier Integral / Transform Explained

Posted by Jeff Schuler on Apr 1, 2011 in Uncategorized |

Magic is real, and it all comes from the Fourier Integral.  But one doesn’t become a wizard without a little reading first – so, the purpose of this article is to explain the Fourier Integral theoretically and mathematically.

Before reading any further, it is important to first understand this: in mathematics, there is a rule that states that any periodic function of time may be “reconstructed” exactly from the summation of an infinite series of harmonic sine-waves.  The generalized theory itself is referred to as a “Fourier Series.”  For use with arbitrary electronic time-domain signals of period T_o, it may be expressed as:

f(t) = a_0 + \displaystyle\sum_{n=1}^{\infty}a_ncos(nw_ot) + b_nsin(nw_ot)

over the range:

t_0 \le t \le t_0 + T_0

where:

a_0 is the magnitude of the 0th harmonic

a_n represents the magnitude of the nth harmonic of cosine wave components

b_n represents the magnitude of the nth harmonic of sine wave components

w_o is the fundamental frequency

t is the variable that represents instances in time

n is the variable that represents the specific harmonic, and is always an integer

This monumental discovery was first announced on December 21, 1807 by historic gentleman Baron Jean-Baptiste-Joseph Fourier.

Joseph Fourier

In order to go from the Fourier Integral to the Fourier Transform, it is necessary to express the previous Fourier Series as a series of ever-lasting exponential functions.  Using an orthogonal basis set of signals described by e^{jnw_ot} of magnitude D_n, we now write the Fourier Series as:

f(t) = \displaystyle\sum_{n=- \infty}^{\infty}D_ne^{jnw_ot}

where j is \sqrt{-1}.

What is the Fourier Integral?

The Fourier Integral, also referred to as Fourier Transform for electronic signals, is a mathematical method of turning any arbitrary function of time into a corresponding function of frequency.  A signal, when transformed into a “function of frequency”, essentially becomes a function that expresses the relative magnitudes of each harmonic of a Fourier Series that would be summed to recreate the original time-domain signal. To see this, observe the following figures:

square-pulse-wave

Figure 1. A Square Wave Pulse, in time

In order to rebuild a square wave with sines and cosines only, it is necessary to determine the magnitudes of each harmonic used in the Fourier Series, or rather, the Fourier Integral (for continuous time-domain signals).  The relative magnitudes of these needed harmonics can be displayed graphically as a function of frequency (widely known as a signal’s frequency spectrum):

sinc-wave

Figure 2. The Fourier Integral, aka Fourier Transform, of a square pulse is a Sinc function. The Sinc function is also known as the Frequency Spectrum of a Square Pulse.

Though the recreation of a signal using an infinite series of sines and cosines is impossible to achieve in the lab, one may get very close.  Close enough that the most advanced lab equipment wouldn’t be able to calculate the error due to tolerance specifications.  This allows engineers to use Fourier Analysis to work with time-domain signals, such as radio signals, television signals, satellite signals and just about any signal you can think of.  By viewing a signal according to what frequency components are contained within it, electrical engineers may concern themselves with magnitude changes in frequency only, and may no longer worry about the signal’s magnitude-changes through time.  Not only is this a very practical concept when working in the lab, it also greatly simplifies the mathematics behind signal conditioning in general.  In fact, the entirety of the Communications industry owes its success to the Fourier Transform for not only antenna design, but a plethora of other applications.

The math behind the Fourier Transform

The derivations that follow have been summarized from Chapter 4 of the textbook “Signal Processing and Linear Systems” by B.P. Lathi, a fine book for students of Communication Systems.

We begin by considering some arbitrary, aperiodic time-domain signal.  An example of this kind of wave would be the output of a microphone after a man speaks a few words into it.  For the actual signal generated by the changes in voltage as the man spoke, we can use Fourier Analysis to describe it as a summation of exponential functions if we instead desire to reconstruct a periodic signal composed of the same voice signal repeating every T_0 seconds.  For an accurate description, it is important that T_0 is long enough such that the repeating arbitrary signals do not overlap.  However, if we let T_0 approach \infty, then this “periodic” signal is simply just the voice signal (or, any general arbitrary function) in time we wanted to describe initially.  Mathematically, we express:

\displaystyle\lim_{T_0\to\infty}f_{T_0}(t) = f(t)

where f(t) is the time-domain function we wish to apply the Fourier Transform on (here, the arbitrary “voice” signal).  For the above equation to be true, f_{T_0} is equal to:

f_{T_0}(t) = \displaystyle\sum_{n=- \infty}^{\infty}D_ne^{jnw_ot}

where:

D_n = \frac{1}{T_0} \int^\frac{T_0}{2}_\frac{-T_0}{2} f_{T_0}(t)e^{-jnw_ot}dt

w_o = \frac{2\pi}{T_o}

It is important to note here that in practice, the shape (aka “envelope”) of a signal’s frequency spectrum is what is of main interest, and the magnitude of the components within the spectrum comes secondary.  This is because amplifiers and other signal-conditioning circuits may be built to alter the magnitude in any way one wishes, and will not affect signal frequencies (so long as the circuits are LTI systems).  Analyzing the envelope of a signal’s Fourier Transform allows one to use intuitive and mathematically-simplified approaches to signal-processing in general, which we shall see later.  For this reason (and also as T_0 approaches \infty) let:

F(w) = \int^{\infty}_{-\infty} f(t)e^{-jw_ot}dt

Notice that F(w) is simply D_n without the constant multiplier \frac{1}{T_0}, such that:

D_n = \frac{1}{T_0}F(nw_o)

which implies that f_{T_0}(t) may be written:

f_{T_0}(t) = \displaystyle\sum_{n=- \infty}^{\infty}\frac{F(nw_o)}{T_0}e^{jnw_ot}

Observation of this fact reveals insight: The shorter the period, T_0, the larger the magnitude of the coefficients.  But, on the other hand, as T_0 \rightarrow \infty, the magnitudes of every frequency component approaches 0 – which is why engineers choose to analyze spectrum envelopes.  So, instead of visualizing absolute frequency magnitudes, instead consider that the frequency spectrum simply expresses the magnitude-density per unit of bandwidth, aka Hz. And since:

T_0 = \frac{2\pi}{w_o}

then:

w_o = \frac{2\pi}{T_0}

and:

\Delta w_o = \frac{2\pi}{\Delta T_0}

so:

f_{T_0}(t) = \displaystyle\sum_{n=- \infty}^{\infty}\frac{F(n\Delta w_o)\Delta w_o}{2\pi}e^{jn\Delta w_ot}

In the limit as T_0 \rightarrow \infty we see:

f(t) = \displaystyle\lim_{T_0\to\infty}f_{T_0}(t) = \frac{1}{2\pi} \int^{\infty}_{-\infty}F(w)e^{jwt}dw

which is referred to as the Fourier IntegralF(w) is referred to as the Fourier Transform of the original aperiodic function f(t), and we express this concept as:

f(t) \Leftrightarrow F(w)

A fourier transform example

This example is from the same textbook as the previous derivation, and can be found on page 239.

Find the Fourier Transform of: e^{-at}u(t) where a is an arbitrary constant.

To do this, we apply the Fourier Integral to the function e^{-at}u(t) as follows:

F(w) = \int^{\infty}_{-\infty}e^{-at}u(t)e^{-jwt}dt

Because of the u(t) factor, we only integrate from 0 \rightarrow \infty.  We simplify for:

F(w) = \int^{\infty}_{0} e^{-(a+jw)t}dt = \frac{-1}{a+jw}e^{-(a+jw)t}\mid^{\infty}_{0}

Also, we know that |e^{-jwt}| =Re[e^{-jwt}] = 1.  So, for a > 0, as t \rightarrow \infty:

e^{-(a+jw)t} = e^{-at}e^{-jwt} = 0

So:

F(w) = \frac{-1}{a+jw}e^{-(a+jw)t}\mid^{\infty}_{0} = 0 - \frac{-1}{a+jw}e^{-(a+jw)0} = \frac{1}{a+jw}

for: a> 0

Useful Fourier Transform Properties

The relationship between f(t) and F(w) exhibit beautiful symmetry that help one to develop an intuitive approach to signal analysis.  Among all the concepts within electrical engineering, the properties between a time-domain function and its Fourier transform are among the most important to understand.  Observe these following properties that apply for all f(t) \Leftrightarrow F(w):

1.) Fourier Transform: Gives an equation to solve for the time-domain function f(t) from F(w).

f(t) = \frac{1}{2\pi} \int^{\infty}_{-\infty}F(w)e^{jwt}dw

2.) Inverse Fourier Transform: Gives an equation to solve for the frequency-domain function F(w) from f(t).

F(w) = \frac{1}{2\pi} \int^{\infty}_{-\infty}f(t)e^{-jwt}dw

3.) Symmetry Property: For a given pair of a time-domain signal and its Fourier transform, we note that the time-domain envelope is different in shape when compared to the frequency-domain envelope.  However, switching the shape of the two functions with respect to domain (time or frequency), will result in the same envelopes except with different scaling coefficients.  For example, a square pulse through time has a frequency spectrum described by a sinc function, and a sinc function through time results in a frequency spectrum described by a square pulse.

F(t) \Leftrightarrow 2\pi f(-w)

4.) Scaling Property: Time-scaling a time-domain signal (by a constant a) will result in a magnitude-and-frequency-scaling of the signal’s corresponding frequency spectrum.  Also signifies that the longer a signal exists through time, the narrower the bandwidth (collection of frequency components needed to rebuild the signal) of its frequency spectrum.

f(at) \Leftrightarrow \frac{1}{|a|}F(\frac{w}{a})

5.) Time-Shifting Property: By time-shifting, or delaying/advancing, a time-domain signal results in a phase delay in each of the ever-lasting frequency-components needed to rebuild it.  The frequency spectrum is otherwise unchanged – only the phase of each component is shifted.

f(t-t_0) \Leftrightarrow F(w)e^{-jwt_0}

6.) Frequency-Shifting Property: Multiplying a time-domain signal by a sinusoidal signal of some frequency w_0, a method which begets amplitude and frequency modulation (AM/FM), results in the frequency spectrum remains unchanged except for a shift in frequency for each individual frequency component by w_0.

f(t)e^{jw_0t} \Leftrightarrow F(w-w_0)

 

Lastly, these tables (table 1, table 2) can greatly simplify Fourier analysis when used in signal processing.

 

 

 


 

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