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
The Fourier transform itself is defined by the equation
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
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
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.