6 Performance

Two types of data were used in the simulations. The first type consisted of software generated signals, in which voice signals were FM modulated, adjusted in power, and added to form a simulated cochannel FM signal. This represents a completely noiseless scenario. The second type of simulation was conducted on signals recorded directly from UHF narrowband FM radios.

Both types of signals were processed on a Pentium 166 running the separation algorithm offline. Parameters used in the program included the sampling rate, decoding delay, order of the Levinson-Durbin linear predictor, and the modulating signal bandwidths of the cochannel signals. The program read in the parameters, computed the linear predictor coefficients from the sampling rate and bandwidths, and then began the Viterbi algorithm as described in the previous section.

There were five test cases for the simulated data set. In each case the the first signal is a male voice and the second signal is a female voice. In all cases, the sampling rate is 132300 Hz., (to match with previous work for the joint Viterbi [7] and cross-coupled phase-locked loop [8]), the SNR is infinity, a fifth order linear predictor is used under the assumption of flat 4 kHz. bandwidth modulating signals, and the decoding delay is 1. The SIR and frequency deviations were varied. Table I gives the normalized mean-squared error (MSE) between the true and estimated instantaneous frequencies, i.e., $\sum_n (\hat\theta'[n] - \theta'[n])^2 / \sum_n \theta'[n]^2$ and $\sum_n (\hat\phi'[n] - \phi'[n])^2 / \sum_n \phi'[n]^2,$ and compares it to the cross-coupled phase-locked loop and the joint Viterbi algorithm. In all cases, both the dominant and subdominant signals were separated perfectly, to within the floating point precision of the computer (normalized MSE of $10^{-20}$ or less), i.e., the correct branch of the trellis was chosen at every step. In addition, the one-step linear predictor itself is very accurate; in every case, the average difference between the linearly predicted phase and the phase given by the chosen state is 1.5 degrees or less. Furthermore, the amplitude estimation is nearly perfect as well, since the signals are noiseless.

Table I: Comparison of cochannel separation methods on simulated data.

Freq.	SIR	Cross-coupled	Joint	Analytical
Dev.	(dB)	PLL	Viterbi	Technique
(kHz)		(MSE)	(MSE)
12	6	0.06/0.48	0.09/0.45	0.00/0.00
12	1	1.28/0.75	0.59/0.66	0.00/0.00
8	6	0.09/0.87	0.16/0.96	0.00/0.00
8	1	1.32/0.79	0.71/0.97	0.00/0.00
12	30	-/-	-/-	0.00/0.00

Each test case in the recorded RF data set consists of a female dominant voice and a male subdominant voice. Here, the sample rate is reduced to 40kHz. Added impairments included noise (SNR = 10-50 dB), Doppler offset (0Hz or 1000Hz), and SIR (6dB, 10dB, or 20dB). The SNR is defined as the ratio of the dominant signal energy to the noise energy.

Fig. 1 shows the normalized mean-squared error of the instantaneous frequency tracking. As a rule of thumb, an MSE less than 1.0 corresponds to an intelligible output. As can be seen, the dominant signal was extracted with excellent quality throughout the SNR range from 10-50dB, and the subdominant signal was intelligible whenever the SNR was higher than the SIR by at least 20dB. The algorithm demonstrates robustness against varying SIR's, and also against varying amplitudes-- the recorded signals varied in amplitude by about 2 dB over the length of the sample. When the amplitudes are assumed to be known, Fig. 1 indicates that the performance does not flatten out as quickly in the high SNR region. A final test in which a Doppler offset of 1kHz was added to the subdominant signal did not appreciably alter the phase-tracking performance shown in Fig. 1.

Figure 1: Instantaneous frequency tracking performance of the analytic cochannel separation technique. The lower set of curves are for the dominant signal; the upper set, for the subdominant signal.

Figure 2: Amplitude tracking performance of the analytic cochannel separation technique.

Fig. 2 shows the normalized mean-squared error of the amplitude tracking using the analytic algorithm when a window length of 5000 samples is used, i.e., $\sum_{i=n}^{i=n-4999} (\hat A[i] - A[i])^2 / \sum_{i=n}^{i=n-4999} A[i]^2$ and $\sum_{i=n}^{i=n-4999} (\hat B[i] - B[i])^2 / \sum_{i=n}^{i=n-4999} B[i]^2.$ As SIR increases, the dominant signal amplitude is easier to track (because of less interference), and the subdominant signal itself is harder to track (because of its lower power).

These results are in stark contrast to the joint Viterbi algorithm and cross-coupled phase-locked loop, which were not able to separate the recorded field data reliably in any SNR or SIR region in Fig. 1. For example, when the joint Viterbi algorithm was used with SIR=6dB and SNR=$\infty$, the resulting phase MSE was 0.34 for the dominant and 1.66 for the subdominant. The cross-coupled phase-locked loop had similar performance.