Fourier Series: Square-wave
The Fourier Series representation of continuous time periodic square wave signal, along with an interpretation of the Fourier series coefficients is presented in this module. This module is meant to bridge the gap between the student and the prescribed text book. Hence material found is most text books is not included here.
The Fourier Series representation of continuous time periodic signals consists of representing the periodic signal as a weighted sum of cosine signals (consider the trigonometric serried or the cosine Fourier series).
Some of the features of this representation are:
2.0 An example:
Let us obtain the Fourier representation of the continuous time periodic square wave given in figure 2. From the above discussion we know that the Fourier series gives the time domain representation of the signal. Before obtaining the Fourier coefficients, let us see if representation without Fourier series is possible.
Figure 1: The continuous time periodic signal
2.1 Representation without Fourier Series:
Consider the continuous time periodic signal of figure 2. It is possible to represent the periodic signal of figure 2, using a set of two equations: (i) the representation in one period, and (ii) the complete signal as a infinite sum of time shifted signals as shown below:
Some observations
Due to the above drawbacks, the Fourier series representation of continuous time periodic signals is preferred.
The Fourier series representation of the periodic signal of figure 2 is given by:
We shall now see the effect of truncating the above infinite series summation.
The one term representation of the periodic signal of figure 2 is given below in figure 3. It can be seen that this is a very poor representation of the original signal.
It can be seen that the power of the signal is 1W, and power contained in the first harmonic if 0.81W, and hence the error in one term approximation is as high as 18.90%
Figure 3: Single term representation of the periodic square wave
The first and the second term in the Fourier series representation of the periodic square-wave is given in figure 4. The sum of these two components results in the two term representation of the periodic square-wave, and is given in figure 5. In this case, the power contained in the first two terms is 0.9W, and hence the error in two term approximation is 10 %.
Figure 4: The first and the second term of the Fourier series of the periodic square wave
Figure 5: The two term representation of the Fourier series of the periodic square wave
The first three terms in the Fourier series representation of the periodic square-wave is given in figure 6. The sum of these three components results in the three term representation of the periodic square-wave, and is given in figure 7. In this case, the power contained in the first three terms is 0.93W, and hence the error in three term approximation is 7 %.
Figure 6: The first three terms of the Fourier series of the periodic square wave
Figure 7: The three term representation of the Fourier series of the periodic square wave
The representation to include upto the eleventh harmonic is given in figure 8. In this case, the power contained in the eleven terms is 0.966W, and hence the error in this case is reduced to 3.4 %.
Figure 8: Fourier representation to contain upto the eleventh harmonic
Since the continuous time periodic signal is the weighted sum of sinusoidal signals, we can obtain the frequency spectrum of the periodic square-wave as shown below in figure 9.
Figure 9: The time-domain and frequency domain representation of the periodic square-wave
Consider the case of the filter input being the square wave and the filter cut-off is such that only the first harmonic gets through, then, the output of the filter will be a sine wave, as shown in figure 10. Suppose we desire the output of the filter to be a fair resemblance of the input periodic signal, then, the filter cut-off needs to be about ten times the fundamental, and in this case the error will be less than 3%. For reduced error, the cut-off could be about 20 times the fundamental frequency.
Figure 10: The output of the low pass filter when cut-off is such that it passes only the first harmonic
We now consider the case of a square wave with period 1 second. The signal and its spectrum is sketched in figure 11(a). In figure 11(b) we have the same signal with period reduced to 0.5 second. From its spectrum it can be seen that the effect of compression in time domain, results in expansion in frequency domain. The converse is true, i.e., expansion in time domain results in compression in frequency domain.
Figure 11: (a) The periodic square wave with period of 1sec, and its corresponding spectrum, (b) The square wave with period reduced to 0.5 second and its corresponding spectrum.