0.7 Appendix a  (Page 3/4)

Implications of the filter design on signal-to-noise ratio and noise power ratio

The pulse response $h\left(k\right)$ chosen for the transmultiplexer determines many of the transmux's key technical performance parameters, including:

• passband bandwidth, gain, and gain ripple
• passband differential group delay (constrained to zero by using an FIR linear phase design approach)
• adjacent channel rejection (also known as intelligible crosstalk )
• channel signal-to-noise ratio (SNR) (also known as unintelligible crosstalk and noise power ratio (NPR), a name based on one method of measurement)

We focus here on the last two. At first it might seem that these two are equivalent, but in fact the second is usually the more demanding of the two. This may be demonstrated by re-examining [link] (a). An adjacent channel rejection specification of 55 dB, say, means that no signal appearing anywhere in the input bandwidth of f _s Hz can appear in the band of interest at any level higher than 55 dB below the signal of interest. If the signal of interest and the one not of interest have the same power levels, then this specification implies that the filter pulse response should suppress the signal not of interest by at least 55 dB before it is potentially aliased into the band of interest. Thus the adjacent channel rejection specification treats each interferer separately and forces each of them to be suppressed to a level below intelligibility. Typical specifications for voice channel demultiplexers, for example, are 55 dB of suppression for any signal more than 300 Hz above or below the channel of interest.

The last specification limits the total noise that enters the band of interest from other channels. These channels are assumed to be statistically independent, implying that whatever energy that aliases into the band of interest from each of the channels is uncorrelated with the others, that their powers add, and that none of them is individually intelligible.

The channel SNR can be quantified by using the expression

where B is the bandwidth of the channel of interest, C is the number of channels with signals present, c is the index of the band of current interest, $H\left(\omega \right)$ is the frequency response of the pulse response $h\left(k\right)$ , and $P_n\left(a\right))$ is the power spectrum of the n -th signal.

If we assume that all C channels have the same average power P , that the power gain of the filter is unity for the channel of interest, and that all other channels are suppressed by exactly S dB, then [link] simplifies to

The implications of [link] can be seen with an example. Suppose that the application at hand requires the demultiplexing of 60 FDM voice channels, that the adjacent channel rejection is specified to be 60 dB, and that the unintelligible crosstalk or SNR specification is 55 dB. Evaluation of [link] shows that α , on which Q and L depend, needs to be about 2.42 to suppress any single component by 60 dB. [link] , however, shows that to satisfy the unintelligible noise specification, all channels not of interest need to be suppressed by an extra $10·lo{g}_{10}59=17.8$ dB For a 960-channel transmultiplexer this extra suppression goes up to 29.8 dB . To meet this specification, the pulse response needs to suppress the other input channels by $55+17.8=72.8$ dB, with the associated growth in α from 2.42 to 2.88 and some additional design concern in hardware using finite word length arithmetic.