# Introduction to Signal Analysis

One looks at various text books on signal analysis and is rapidly confronted with calculus. Hopefully it will bring a sigh of relief since your scribe is not conversant with that area of maths, and his text editor is not capable of producing such functions in print.

Rather, this is a layman's guide to some of the terms used in signal analysis and an introduction to understanding those terms and parameters mentioned in currently available software packages.

## Fourier Transformations

Jean Baptiste Joseph, a Frenchman, developed the Fourier Series which shows how repetitive pulse signals can be expressed as a set of sinusoidal frequencies (fundamental + harmonics).

From this came the mathematical tool known as the Fourier Transform. It takes a signal from the time domain (ie amplitude v time) and converts it to its frequency domain (ie amplitude v frequency). A frequency domain plot is also known as a spectra plot or spectrum.

The basic Fourier Transform is often referred to as a Discrete Fourier Transform (DFT); discrete because it depends on discrete instants in time. These instants are the times the software samples the input signal. However being based on discrete samples does not mean a lessening of any essential detail from the original signal.

The FFT (Fast Fourier Transform) was developed in 1965 (Cooley+Tukey). It had been shown that many of the multiplication operations in the DFT were repeated. By a means of algorithms FFT tries to reduce these repetitions thereby increasing calculation efficiency, particularly as the number of samples increases. Calculation is said to be more efficient where it is carried out on a block of samples where the block size is an integer power of two eg 1024 or 2048.

## Investigating the Spectrum

One of the aspects of monitoring is investigation of the frequency spectrum associated with the signals being heard. Programs like Spectralab and Spectrogram are available which allows one (with varying degrees of flexibility) to analysis the audio spectrum being produced by the receiver. These notes are intended to outline the various primary parameters one is likely to come across.

For the purposes of this exercise Spectralab is utilised. It should be appreciated that these programs are processing some complex mathematical equations (number crunching) and will require the provision of computer power to the minima given in the respective documentation to be able to cope with all eventualities/settings especially when running in real-time.

A sprectrum is a 2 dimensional view where the horizontal axis is the frequency (Hz) and the vertical axis shows the relative amplitudes.

## Sampling Rate

A gentleman by the name of Nyquist determined that, in order to provide sufficient sampling to correctly handle the maximum desired frequency, the sampling rate had to be twice that frequency. Amongst others SpectraLAB offers 5000 and 8000 samples/sec. This permits maximum values of 2500 and 4000 Hz respectively.

If one is dealing with signals ex a receiver, and not a hi-fi system, the AF is limited by the IF bandwidth (typ. 2.4-2.6 kHz). Some higher grade (or ex professional) receivers may have wider filters say 3.0-3.2 kHz for dealing with wide vft's.

So 8000 is adequate. No purpose in going higher. It should be noted that if the sampling rate is raised the resolution becomes coarser. If lowered the resolution is finer but we have already determined the lower limit for the sampling rate.

## Sampling Precision

This can be set to use 8 or 16 bits. When the analog signal is sampled it is converted or digitised into a digital form with the coding in an 8 or 16 bit form. The latter gives finer resolution and therefore desirable but requires more resource in computer power/processing time.

## FFT Size

This is the number of samples in the block on which the current calculations are being carried out. This should be set initially to 1024.

Note that increasing this value increases the resolution of the plot giving a smaller width between spectral lines but this will increase the time to carry out the calculations

## Smoothing Windows

The spectrum produced by the FFT will be correct (i.e. a each spectral line with correct amplitude/frequency) provided the sine wave being sampled is crossing zero (amplitude at the start/end of the time series.

However a discontinuity occurs (due to the sample waveform being truncated) should sampling occur off-zero. The FFT therefore produces smearing onto the neighbouring lines. This is termed leakage.

The practical way of reducing this effect is by use of smoothing windows.

There are a number of types available. They have different shapes which modify the weighting between the edges/centre of the sequence. If there is no windowing function it may be referred to as Flat, Rectangular or Uniform.

As will all things in life compromise is the order of the day and windowing is no exception being a balancing act between frequency resolution (FR), amplitude resolution (AR) and leakage suppression (LS) such that an improvement in one characteristic results in the deterioration of another.

For radio digital monitoring purposes it would seem the three best suited would be:

```Type            FR          AR          LS

Flat(Uniform)   Exclnt      Poor        Poor
Hanning         Fair        Exclnt      Exclnt
Blackman        Fair        Good        Exclnt

```

with Flat being the entry choice.

The displayed spectrum can be further smoothed by averaging the previous x computed traces where x is the AVERAGING (BLOCK SIZE) parameter value.

If the signal being analysed rapidly changes in frequency a low value should be used but one may find this somewhat disjointed. On the other hand a high value is recommended to enhance a weak steady signal (carrier) in a noisy environment.

A useful start point has been found to be 30. If averaging options are provided, Linear should be chosen being known as stable averaging.

Spectrogram is a more entry level piece of software and does not have Spectralab's flexibility but real time analysing is possible. This is enabled by selecting FILE/Scan Input or [F3].

This brings up a dialog box allowing selection of parameters including Sample rate (5.5K will cover audio to 2750hz), sample resolution (8/16bit), Frequency scale (linear), FFT size and spectrum averaging. However to change paramters one must stop the strace and press [F3] again. There is no smoothing window type selection in this package.

In both Spectralab and Spectrogram the display's horizontal range may be adjusted (within limitations set by FFT size). In Spectralab the lower and upper ends can be adjusted independently. 300 Hz and 3000 Hz (approx) are useful values to match the output from a monitor's receiver. Spectrogram has a fixed range which must be adjusted to match the area of interest ie no independant adjustment of ends.

Finally, as with decoders, take care not to overload the input as this will cause distortion and incorrect readings. While the suggested parameters are not the be-all and end-all they are given as useful start points. Like every other thing in life it is possible to become confused by incorrect setting of many variables - develop skills by taking a good signal and moving individual parameters either way and noting the effect. Return that one to its "datum" and try another. This way one hopefully learns not to induce variations in the trace that aren't in the signal. At the end of the day one may, by personal experience, find one's own datum set of values - each to his own.

From the WUN0508 Digital Review by Day Watson pub. Aug 1999 (c) Worldwide Utility News
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