Question

When exploring the effect of changing the Fourier coefficients on creation of specific waveforms, answer the following questions about square waves

1. Why are the components only sine waves and not cosines?

2. Why do you only use every other wavelength of sine waves?

3. What happens to the result if you do not decrease the amplitude of the waves as you go to shorter and shorter wavelength? (Try setting them to the same size and see what happens.)

Answer 1

The required wave form is finite in a specified range and is zero outside the range. This is obtained by adding an infinite number of harmonic integrals of continuously varying amplitudes and frequencies.

This is Fourier integral , a(k) gives the amplitude of each harmonic wave with wave number k .

e^ikx = Cos kx + i Sin kx

we need the pulse exist within a specified range -T to +T

In the integral the sine function is an odd function and gets canceled. The integral reduces to only Cos function and the result of the integral is a Sin function and hence the components are only Sine waves

b) we use both +ve and -ve frequencies spread uniformly around a main frequency

c) the amplitude component a(k) can be constructed if we know f(x)

as

The amplitude decreases with higher frequencies since the pulse must vanish beyond a specified range. If the amplitude does not decrease the resultant pulse or the signal is of infinite range , which is not the case in real world.