# Pulsed Signals Spectrum Analysis

## How to measure pulsed signals using a spectrum analyzer – some of the key concepts and techniques.

Pulsed signals are used in a variety of areas of electronics and radio. One particular example is within radar signals, but many other applications exist.

In view of their use, it is often necessary to use a spectrum analyzer to measure the characteristics of the signals. However the fact that they are pulsed raises some interesting challenges.

Traditionally pulse spectrum analysis techniques and approaches are normally aimed at steady analogue RF signals. However pulse spectrum analysis demands a little understanding of the signals being analysed, and this can enable additional information to be gained.

## Pulse signal spectrum analysis basics

Radio frequency pulsed signals, or any form of pulsed signals take a variety of forms, but despite the variety, they have a number of common traits. This means that it is possible to apply common pulse spectrum analysis techniques.

To look at the techniques used for pulsed signal spectrum analyzis, it ids first necessary to look at the basic nature of a pulse waveform. It has a repetition time of T and the pulse duration of t.

Using Fourier analysis it can be seen that this waveform is made up from a fundamental and harmonics. The basic waveform of a square wave can be made up from the fundamental sine wave with the same repetition rate as the square wave and then odd harmonics with the amplitudes of the harmonics inversely to their number.

A rectangular pulse is just an extension of this basic principle. The different waveform shape is obtained by changing the relative amplitudes and phases of harmonics, both odd and even.

These base-band signals can then be plotted and the amplitudes and phases of the infinite number of harmonics, both odd and even result in the smooth envelope shown below.

The envelope of this plot follows a function of the basic form:

$\gamma =\frac{\mathrm{sin}\left(x\right)}{x}$

This single can then be modulated onto an RF waveform to give a spectrum. As the harmonics of the baseband signal, extend out to infinity, so too do the sidebands of the modulated signal. In reality, however, the bandwidth will never be infinite and the harmonics, especially higher order ones are attenuated. Although this results in some distortion of the signal, the levels are generally acceptable.

## Pulse spectrum analysis

It has been possible to see how pulse signals are generated and the resulting spectra. While the phase of the sidebands is accommodated on the plots above, spectrum analysers are scalar test instruments and do not normally give an indication of the phase of a signal. Accordingly the plots from spectrum analysers are only shown "above the line.".

There are a number of points can be noted for this:

• Spectra lines:   The individual spectra lines shown on the graph of the modulated waveform are separated by a frequency equal to 1/T.
• Nulls in envelope:   The nulls in the envelope or overall shape of the spectra occur at intervals of 1/t. Further nulls occur at n / t
• Envelope null distinctness:   The nulls in the pulse spectrum shape are not always particularly distinct because of the finite rise and fall times in the modulating signals and the resulting asymmetries that exist.

## Pulse desensitisation

Sometimes the issue of pulse desensitisation is referred to in terms of pulse spectrum analysis. The issue is that when the modulation is applied to the carrier, the peak level of the envelope is reduced, appearing that the signal has been reduced in overall power.

The apparent reduction in peak amplitude occurs because adding the pulse to the signal and modulating it with a square wave results in the power being distributed between the carrier and the sidebands. As the level of the modulation increases, so does the level of the sidebands. As there is only limited power available and each of the spectral components, i.e. carrier and sidebands, then contains only a fraction of the total power.

The overall effect as seen on a spectrum analyser is that the peak power reduces, but it is spread over a wider bandwidth.

It is possible to define a pulse desensitisation factor α. This can be described in the equation:

$a\left[dB\right]=20{\mathrm{log}}_{10}\left(\frac{{t}_{eff}}{T}\right)$

It should be noted that this relationship is only really valid for a true Fourier line spectrum. For this to be applicable the resolution bandwidth of the analyser should be < 0.3 PRF.

The average power of the signal is also dependent on the duty cycle as the power can only be radiated when the signal is in what may be loosely termed the "ON" condition. This can be defined by the equation below:

Where:
α = Pulse "desenitisation factor
T = pulse repetition rate
PRF = Pulse Repetition Frequency (1 / T)
t = pulse length
teff = effective pulse length taking account of rise and fall times
Pavg = Average power over a pulse cycle
Ppeak = Peak power

## Triangular and trapezoidal waveforms

While pulse spectrum analysis is normally applied to square or rectangular waveforms, similar principles also apply to triangular and trapezoidal waveforms.

The format of the waveform has many similar characteristics to those of a pulse waveform but with different levels of the different constituent signals and hence the sidebands.

It is therefore possible to analyse these waveforms in a similar way.

## Pulse spectrum analysis measurement tips

When looking at a pulsed signal using a spectrum analyser it is necessary to employ techniques to ensure that the signal is displayed to reveal the aspects that are required.

Some of the chief aspects are:

• Measurement bandwidth less than line spacing:   To resolve the individual spectral lines, the measurement bandwidth must be small relative to the offset of the lines, i.e. Bandwidth < 1 / T. If the measurement bandwidth is reduced further, them the spectral lines will retain their value (as expected) but the noise level will be reduced, although measurement time will be longer.
• Measurement bandwidth between line spacing and null spacing :   The next stage occurs when the measurement bandwidth is greater than the spectral line spacing, but less than the null spacing. For this condition the spectral lines are not resolved and the amplitude height of the envelope depends upon the bandwidth. This is because a greater number of spectral lines, each with their own power contribution re contained within the measurement bandwidth. For this case 1 / t > B > 1 / T.
• Measurement bandwidth greater than null spacing:   For this case where the measurement bandwidth is greater than the null spacings on the signal spectrum envelope, i.e. B > 1 / T, the amplitude distribution of the signal cannot be recognised.

With pulse transmission being widely used, pulse spectrum analysis is an important element of characterising and testing any equipment that is developed and the signals they produce.