# SINR versus transmission rate

In the following paper on applications of Game Theory to power control in communications,

Why is the Signal to Interference and Noise Ratio (SINR) inversely proportional to the transmission rate R?

The expression for the SINR can be found in section III of the paper.

Thanks

The performance of a communication system depends to a great extent on E_b, the amount of energy received per bit. This energy is the integral of the instantaneous (received) power per bit, or, if you like, the average power over the bit duration times the bit duration. So, if the bit rate is increases, the bit duration decreases and so does E_b. Since SNR or SINR is the ratio of the bit energy to the noise or interference-plus-signal energy, the SNR and SINR decrease as the bit rate increases.

If you like to think in terms of power and bandwidth, and think of SNR or SINR as the ratio of signal power to noise power etc., then consider that decreasing the bit duration means using shorter pulses, and so more bandwidth is needed. As a consequence, there is more noise in the receiver since it needs to have larger bandwidth to capture and process the wider-band signals. Thus, while signal power remains fixed as the bit duration decreases, the noise power increases, leading to the same conclusion as before:

As bit rate increases, SNR and SINR decrease.

• can you please clarify why: decreasing the bit duration means (using shorter pulses, and so) more bandwidth is needed?. I fail to visualize that in my head... Commented May 19, 2015 at 17:41
• @KristofTak Consider the Fourier spectrum of a pulse of short duration versus a (similar) pulse of longer duration. Which has more content at high frequencies? If the channel does not have enough bandwidth, the short pulse will be smeared out in time and give rise to inter-symbol interference which brings in additional complications that need to be taken into account. Commented May 19, 2015 at 18:49

If your power is fixed, then as you increase your bit rate, the energy per bit interval is decreased at the same rate. You can think of this as integrating the power over the length of the bit. If you integrate over less time, your energy for that bit will also be less. Specifically, if you double your rate, you cut your energy per bit in half. This is why the SINR is inversely proportional to the rate.

• I think there is some confusion here with power per bit interval being used in lieu of the term bit energy, the integral of the power over the bit duration. Doubling the bit rate does not cut the power per bit in half, it cuts the energy per bit in half. A 50-watt light bulb uses 50 watt-hours per hour; if you use it for 30 minutes, it is still a 50-watt bulb, not a 25-watt bulb. Power and energy mean different things.... and power per bit is an abuse of nomenclature that is best avoided. Commented Feb 16, 2012 at 21:32
• @DilipSarwate Thanks for the catch, I answered in a rush. Commented Feb 16, 2012 at 21:43