Instantaneous Frequency Plots and AM

All of the experiments so far have dealt with the analysis of a single filter channel, although this allows us to examine the effects of AM in very fine detail it is sometimes necessary to take a step back and have a look at the effects of AM over a larger frequency/time area.

One way of examining a more general view of the waveform is to use a time/frequency spectrogram, the usual method of filter bank analysis. However, this will only give information on one variable at a particular point in time/frequency. This is variable is usually energy, however, we wish to display information relevent to the AM of the signal, such as changes in IF as well as energy. It is necessary therefore to encode more than one variable onto each point of time/frequency.

There are two main methods of encoding more data onto a single x/y point, the first is to either extend the dimensionality, e.g. have x,y, and z. Unfortunately when displaying more than 2 dimensions on a screen is tricky because the high level data always tends to obsure the lower level data. The other method is more useful to us though, that of encoding extra variables by using colour. Colour allows the encoding of an extra two variables on each pixel, represented by the hue and saturation.

This type of colour representation is known as HSV or Hue, Saturation, and Value. Hue is a representation of 'colour' based on a 360 degree colour wheel where red, green, and blue combinations are spaced around the wheel. Saturation is the amount of colour, where 1 represents a deep colour, whilst a value of 0 would show as white. Finally, Value rated from 0 to 1is the same as that used in monochrome spectrograms, that of brightness.

By using HSV with a normal spectrogram we can use the Hue and Saturation variables to show the changes in IF seen in the previous section. Differences between the IF and the centre frequency of a filter bank channel are represented by colour saturation, with the value (or colour) showing blue if the IF is above the filter CF and red when below. Using this system the spectrogram will show the dominance of frequencies within filter channels, with channels containing frequencies close to their CF being shown as white, changing to a deeper and deeper colour as the difference between the CF and the IF grows.

An example of this representation can be seen below in Figure's 1 and 2, these show spectrograms of the result of adding two frequencies, one at 1100Hz, the other at 900Hz. In Figure 1 the 1100Hz tone has a higher amplitude than the 900Hz tone, in Figure 2 the reverse is true.

Figure's 1 & 2 : Combined Tones of 1100Hz and 900 Hz

As can be seen, the frequencies of both source tones are easy to spot as the two white lines with the higher amplitude tone showing brightest. The effects of the intereference are of course greatest somewhere between the tones where they both have equal amplitude. It is here that we see the period of beating is most obvious by the appearance black 'holes' where the envelope is reduced to zero by wave interference. Looking across time near these areas we see that between the holes we have areas of very weak colour, indicating that the IF here is very near the filter cf. This shows the behaviour explained in ' AM Theory for Tones with Unequal Amplitude ' where the frequency of the AM signal is shown to be a weighted average of the two tones, therefore always near the filter CF. However, as the envelope of the waveform reduces the IF deviates (Delta F) away from the CF, as shown by the stronger colours as the 'holes' are approached. In fact if there where any energy at these points this is where the colours would be strongest, since this is where the IF deviation 'spikes' occur.

Analysis of AM in Formants

Using this display technique it is possible to examine the effects of AM when adding signals with formants from sources with different fundamental frequencies. Using the Klatt synthesiser to create speech signals with F1 and F0 only, generated with different pitch it is possible to see how wave interference affects speech when speakers formants are similar in frequency. In order to examine this effect 7 different signals were generated. A reference signal, with 100Hz F0 and 300Hz F1, was combined with each of the 4 test signals. The test signal's F0 and F1 frequencies were as follows:

Signal# F0 Freq F1 Freq
1105Hz300Hz
2110Hz300Hz
3120Hz300Hz
4120Hz350Hz

These signals were then combined and analysed using 256 filter channels from approximately 250Hz to 400Hz, the images produced from this analysis can be seen below in Figure's 3-6:

Figure 3: Reference Signal with Test Signal 1, Figure 4: Reference Signal with Test Signal 2

Figure 5: Reference Signal with Test Signal 3, Figure 6: Reference Signal with Test Signal 4

As can be seen, Figure's 3-5 show the test signal combined with the 300Hz formant show strong beating due to the Amplitude Modulation caused by the two signals. This is because whilst the F1's were both at 300Hz, they had a different pitch, therefore the beating frequency is due to the difference of the two pitch frequencies. As the difference between pitch's doubles with each test signal (5Hz,10Hz,20Hz) we see a corresponding inverse change in the period of the AM beating, i.e. as the pitch difference doubles the beating period halves. Finally, Figure 6 shows the combination of a 300Hz and 350Hz F1, this is more like the images seen in Figure's 1 and 2 with the same 'holes' found between 300 and 350Hz. However, instead of beating at a frequency of the difference of the two tones (25Hz) the beating is still at the same frequency of Figure 5, relevent to the pitch difference of the two tones.

In the next two images noise was added to the signal shown in Figure 6 in order to test how robust the IF deviations caused by AM are to noise. The images below show the same signal but with white noise added at an equal level to the signal, and at twice that of the signal:

Figure 7: S/N Ratio = 1, Figure 8: S/N Ratio = 0.5

As can be seen, the regular pattern of IF changes due to the AM of the two formants seen very clearly in Figure 6 starts to break up quite significantly when noise is added. When the level of noise in equal to that of the signal, as shown in Figure 7 you can already see the rounded structure of IF departure around the waveform envelope troughs deforming. When the noise reaches twice the level of the signal the corruption has significantly increased, so that the AM 'troughs' no longer show so obviously.


These analyses show that the interaction of adjacent signals shows quite significantly in IF deviations across both time and frequency, especially in the region of an envelope trough. In the next section,' Analysis of IF Differences over Time and Frequency ', the approaches used for the analysis of IF will be expanded upon in order to give these areas further investigation

Modified 4/4/96 by Jeremy Goslin
 
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