Homework Sourse2 The Fourier transform of the following signal xt is given
Solution
Fourier Series
The Fourier series is concerned with periodic waves. The periodic wave may be rectangular, triangular, saw tooth or any other periodic form(single valued). Here I will also call this periodic wave as signal wave.Any periodic signal wave can be represented as a sum of a series of sinusoidal waves of different frequencies and amplitudes.
Otherwise we can also say that a series of sinusoidal waves of different frequencies and amplitudes add up to give a periodic wave of non-sinusoidal form.
Let us consider a periodic signal wave v . According to the definition of Fourier series, this periodic signal wave can be written as sum of sinusoidal waves as below.
Now the question is how can we find the coefficients
I am not going to show you the details of how I obtained the formulas, but you can remember the three general formulas as shown below for obtaining the coefficients. Many of you may not require to find the harmonics by using the formulas below, but those who are interested can use these formulas. Of course, students are required to remember these formulas.
n = 1, 2, 3, 4 .....
Putting the values of n we obtain different coefficients. For example if n=1 we get a1 and b1.
Some knowledge about the properties of the Fourier series will immensely help you. In most cases signal waves maintain symmetry. Depending on the symmetry of the wave we may not be always required to find all the sine and cosine terms coefficients. So now I will guide you through some important properties that you should remember so that just looking at the signal wave you can immediately say which coefficients should be present in the series.
Let us first of all talk a little about harmonics.
What is harmonic ?
Every periodic wave has a Time period(T) which is one complete cycle. The whole signal is the repetition of this period. Frequency (N) of the wave is the number of complete cycles in one second.
clearly N = 1 / T
Observe the general Fourier series, it has a component (a1 sin wt), this sinusoid has the same frequency as the actual signal wave and it is called the fundamental component. The next component is (a2 sin 2wt), its frequency is twice that of fundamental (or original signal wave), this sinusoid is called second harmonic. So also the third, fourth etc.
As an example if the fundamental wave has a frequency of 60 Hz, then the frequency of second harmonic is 120 (2* 60) Hz, third harmonic 180 Hz, 9th harmonic will be 540 Hz etc. The frequency of nth harmonic is (n.60).
Properties of Fourier Series
We will now consider some important and very useful properties of Fourier series
Looking at the figure it is clear that area bounded by the Square wave above and below t-axis are
A1 and A2 respectively. Here A1=A2, so the average is zero. The wave has a0 equal to zero. You
can also confirm by using the formula above. But shifting the same wave in vertical direction as
redrawn in Fig-B will automatically introduce a0 in Fourier series.
Yes by integrating using the formula above you will find a0. By shifting I have actually disturbed the
symmetry around horizontal axis and introduced the average value a0. Here in this figure a0 is exactly how
much I shifted the whole wave in vertical direction. This a0 is also called DC component of the signal
wave.
We know that if, p = a1 sin x +b1 cos x then it can be written in the form
p = r sin (x + phi) phi is the angle displacement from sin x.
Using school trigonometry r & phi are found from a1 and b1
(Remember that same harmonics are combined in this form. sin x with cos 2x or cos x with sin 2x etc.
are not combined in above form).
As the above square wave maintains symmetry about origin so it will be composed of sine waves only.
(sine waves are symmetric about origin). In Fig-C, I have reproduced the square wave. Here I have
calculated only three terms of the Fourier series \' v \', the fundamental, third harmonic and fifth harmonic.
The blue curve is the sum of these three terms of the Fourier series of the square wave shown. You can
see how just by considering three terms of the Fourier series, the blue curve approximate the square
wave. In Fig-D the sum of Fourier series(in blue) is drawn by calculating more coefficients( 9 ). Observe
how the blue curve approximates the suare wave so that I had to remove the square wave from the figure for clarity. Taking more terms the curve will be even smoother.
The Fourier series is:
It is very important to compare Figure-C and Figure-E. In both cases the square waves are identical.
Only the vertical axes are chosen differently in each cases. Comparing the individual harmonics and their
sum (blue) in both the figures it is clear that actually choosing the position of vertical axis does not
change the component harmonics but what previously you were getting as sine waves have now
become cosine waves automatically. This is only due to the choice of vertical axis.
Both are sinusoidal waves and both are correct. If some intermediate position of axis is chosen, we will
get both sine and cosine terms. The sine and cosine terms can be combined as above to get only either
phase shifted sine(or cosine) terms for each harmonic.
So the choice of axis does not change the shape or number of harmonics, only mathematically the series
is different.
So it is important to realize that for a particular shape of signal wave the harmonics are fixed, whether
you want to show them as series of sine terms or cosine terms or both, it does not matter. Of course in
exam you may not be allowed to choose axis.
Note: A sine wave is symmetric with respect to origin (odd function) and cosine is symmetric with
respect to y-axis (even function). Combination of only sine waves will be symmetric with respect
to origin (odd function) and combination of cosine waves will remain symmetric with respect to
y-axis (even function)
