Fourier Transforms

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Fourier Transforms

The Fourier transform defines a relationship between a signal in the
time domain and its representation in the frequency domain. Being a
transform, no information is created or lost in the process, so the
original signal can be recovered from knowing the Fourier transform,
and vice versa.

The Fourier transform of a signal is a continuous complex valued
signal capable of representing real valued or complex valued
continuous time signals.

The tool allows you to view these complex valued
signals as either their real and quadrature (also known as imaginary)
components separately, or by a magnitude and phase
representation. You may switch between these two representations at
any point. Mathematically switching between the two representations
for a given complex value can be expressed as

and

or equivalently,

and

where

and

are the magnitude and phase of the complex number, and

and

are the real and quadrature components of the complex number. In this
tool, the magnitude is plotted on a dB scale. Select a few
signals, such as unit pulses and sine waves, and view them using the
two methods to see how they are related.

The Fourier transform itself is defined by the equation

where

is the Fourier transform of

Frequency is measured in Hertz, with

as the frequency variable.

Fourier transform of signals

Using the tool, display the Fourier transform of a 4ms unit
pulse. You will observe that the frequency response is a continuous
signal with a maximum at 0 Hz, and some periodicity. The frequency
response is zero at every multiple of 250Hz. Compare this with the
frequency response of a unit pulse of 8ms in duration. Here the
general shape of the signal is the same, but the zero crossings are at
a spacing of 125Hz. These figures are the reciprocals of the pulse
duration, indicating that there are inverse relationships between
time and frequency. Generally, longer time periods relate to smaller
frequency spans.

The formula for the frequency response of a unit pulse may
be calculated directly from the Fourier transform equation as

where

is the duration of the pulse. You can observe the changes in magnitude
caused by the different values of

, as well as the changes in the spacing of the zero crossings, a
function of the sin component.

Sinusoids and cosinusoids are signals that by definition contain only
one frequency of signal. The tool has two examples of these
with frequencies 333Hz and 500Hz. The time domain and frequency
transform of a 500Hz cosine wave is given by the following
equations

Delaying a 500Hz cosine wave by 0.5ms results in a sine wave signal,
and its transform can be seen to be

As this change is made, by adding the delay, you will observe that the
phase of the frequency transform changes, but the magnitude remains
the same. Alternatively, using the real and quadrature representation,
components that were purely real before becoming imaginary after the
delay.

Delay and phase change

Any of the signals can be advanced or delayed by a number of
predefined delays of up to 4ms. Alternatively, you can delay a signal
by an arbitrary amount by clicking and dragging the graph whilst
holding down a key on the keyboard. In the frequency domain this
relates to alteration of the phase of the signal, thus no difference
will be observed when viewing the Fourier transform magnitude plot,
but will be evident when viewing the phase of the transform or the
real and imaginary parts together.

Try this out for various types of signal.

Take particular note of the scaled unit impulse as without any delay
it results in a purely real transform of height 0.004 (the scaling
factor). When this signal is delayed, the transform becomes a
cosinusoid in the real component and a sinusoid in the imaginary. The
formula for this is

where is the
value of the delay.

This implies that a delay of a specific amount in the time domain
equates to multiplication by a phasor in the frequency domain. Set
the delay for the scaled unit impulse to 0.5ms as was done for the
500Hz cosine waveform in the previous section. Now note the values of
the real and imaginary parts of the transform at 500Hz and -500Hz. Now
switch the input signal to the 500Hz cosine and you should be able to
explain how the purely real transform of the undelayed waveform
relates to the purely imaginary transform of the delayed
signal.

Not only can the time domain signal be delayed, but the frequency
transform can be shifted, resulting in a phase change in the time
domain. Experiment with this observing the time domain signal as
magnitude and phase, and as real and quadrature to see the effects
that can be obtained. Try shifting the frequency response of a
cosinusoid, or a sinusoid, so that one of the frequency samples is set
to 0Hz. The result will be a complex phasor, consisting of a
cosinusoid and sinusoid in the real and imaginary components of the
time domain plus a DC offset from the 0Hz component.

Multiplication and convolution

Using the tool, review the transforms of the unit
pulse function and the cosine function. For the moment it is best to
view these using the magnitude and phase representation of the
frequency domain.

Now switch to one of the 8ms segment of a cosine or sine
waveforms. You should observe that the frequency domain plot is some
form of combination of the two types of signal. Strictly speaking, the
time domain signal is the multiplication of a unit pulse of 8ms
duration delayed by 4ms, and a cosinusoid or sinusoid waveform of the
selected frequency. The frequency domain transform is then the
addition of two sa functions which have been shifted in
frequency. Notice where the highest peaks are and you should observe
that these correspond with the frequency of the sine or cosine
signal. What has happened is that in the frequency domain the sa
function from the unit pulse and the two impulses from the sine or
cosine function have been convolved together. This is an example of
the general rule that multiplication in the time domain equates to
convolution in the frequency domain.

You can reconstruct the two constituent waveforms by shifting the
frequency response of the 8ms unit pulse to 500Hz, and to -500Hz.You
should find that the real component of the two shifted signals are the
same, but that the quadrature components are the complement of each
other. Thus when they are summed together, the result is a signal with
a real component and a zero quadrature component.

In fact an equivalent rule also holds that
convolution in the time domain equates to multiplication in the
frequency domain. Thus, for example, a complex phasor in the frequency
domain multiplied by a given signal's transform produces a time domain
function where an impulse is convolved with signal. This is precisely
what is happening when the delay value is being altered.

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