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2.7 The Spectrum of an Ideal Square Wave

An ideal square wave has a zero rise time, by definition. It is not a real waveform; it is an approximation to the real world. However, useful insight can be gained by looking at the spectrum of an ideal square wave and using this to evaluate real waveforms later. An ideal square wave has a 50% duty cycle, is symmetrical, and has a peak voltage of 1 v. This is illustrated in Figure 2-6.

Figure 2-6

Figure 2-6 Time and frequency domain views of an ideal square wave.

If the ideal square-wave repeat frequency is 1 GHz, the sine-wave frequency values in its spectrum will be multiples of 1 GHz. We expect to see components at f = 1 GHz, 2 GHz, 3 GHz, and so on. But what are the amplitudes of each sine wave? The only way to determine this is to perform a DFT on the ideal square wave. Luckily, it is possible to calculate the DFT exactly for this special case of an ideal square wave. The result is relatively simple.

The amplitudes of all the even harmonics (e.g., 2 GHz, 4 GHz, 6 GHz) are all zero. It is only odd harmonics that have values. The amplitudes, An, of the odd harmonics are given by:

Equation 2-3

Chapter 02 Equation 03

where:

  • An = the amplitude of the nth harmonic
  • p = the constant, 3.14159...
  • n = the harmonic number, only odd allowed

For example, an ideal square wave with 50% duty-cycle and 0 v to 1 v transition has a first harmonic amplitude of 0.63 v. The amplitude of the third harmonic is 0.21 v. We can even calculate the amplitude of the 1001st harmonic. It is 0.00063 v. It is important to note that the amplitudes of higher sine-wave-frequency components decrease with 1/f.

If the transition-voltage range of the ideal square wave were to double to 0 v to 2 v, the amplitudes of each harmonic would double as well.

There is one other special frequency value, 0 Hz. Since sine waves are all centered about zero, any combination of sine waves can only describe waveforms in the time domain that are centered about zero. To allow a DC offset, or a nonzero average value, the DC component is stored in the zero-frequency value. This is sometimes called the zeroth harmonic. Its amplitude is equal to the average value of the signal. In the case of the 50% duty-cycle square wave, the zeroth harmonic is 0.5 v.

To summarize:

  • The collection of sine-wave-frequency components and their amplitudes is called the spectrum. Each component is called a harmonic.
  • The zeroth harmonic is the DC value.
  • For the special case of a 50% duty-cycle ideal square wave, the even harmonics have an amplitude of zero.
  • The amplitude of any harmonic can be calculated as 2/(p x n).
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