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If we could transmit a 1 kHz sine wave over a channel having a bandwidth of 1 kHz, how much bandwidth do we need if we would like to transmit a 1 kHz square wave?The Fourier Series is named after the French mathematician Joseph Fourier. Fourier Series has certainly been one of the most must have tool in Signal Processing.
It is so important but today I’m just going to illustrate one aspect. Let me begin with a question.

If we could transmit a 1 kHz sine wave over a channel having a bandwidth of 1 kHz, how much bandwidth do we need if we would like to transmit a 1 kHz square wave?

The answer if infinity! At least this is true for a perfect square wave. To understand this we will need to look into Fourier Series.

As you all know that virtually any physical waveform can, in fact, be represented as the sum of a series of sine waves.

The square wave is no exception, and can be approximated by the expression:

Above figure tells us we need infinite bandwidth to transmit a square wave!

However, a perfect square wave is discontinous; the change from the low state to the high state occurs in zero time.

Any physical system will requre some time to change state. Therefore, any attempt to transmit a square wave must involve a compromise.

In practice, 10 to 15 times the fundamental frequency provides enough bandwidth to transmit a high-quality square wave. Thus to transmit our 1 kHz square wave would require something like 10 kHz bandwidth channel.

A wider channel would give a sharper signal, while a narrower channel would give a rounded square wave. We can say, the higher the frequency that a system can handle, the faster it can change value, and vise versa.