# PSK Bandwidth for SNR

I'm conducting experiments on several n-ary Psk signals of differing symbol rates. I'm comparing detection levels of different signal detection approaches on sets of PSK signals. I'm using a cosine-roll-off filter with a roll-off-factor of between 0.2 to 0.5.

To determine the appropriate signal to noise ratios, I had to decide which bandwidth I was gonna use. For reasons of practicality, I'd like to use the approximation symbolrate[bd] equals bandwidth[Hz]. It was proposed in chapter 6.2.4 of the book:

Communications and Information Systems; Michael John Ryan,Michael Frater

Do I act reasonably and doesn't it skew my evaluation with inappropriate error levels?

• what pulse shaper are you using? – Marcus Müller Apr 11 '17 at 9:49
• Dear potential editor: I rejected the edit. There was no indication whatsoever that Marcel had to deal with symbol rates in dB (how do you dB a rate? It's not unitless, so you'd have to relate it to some fixed frequency), and when TeXing formulas, please do use good TeX practice (e.g. $f_S$ instead of $Fs$; the latter looks like a product or so). – Marcus Müller Apr 11 '17 at 10:31
• I'm using a cosine-roll-off filter with a roll-off-factor of between 0.2 to 0.5. – Marcel Apr 11 '17 at 10:52
• added that to your question :) – Marcus Müller Apr 11 '17 at 10:54

I'm using a cosine-roll-off filter with a roll-off-factor of between 0.2 to 0.5.

Shamelessly copying from wikipedia's Raised-Cosine page, here's a plot of frequency responses of Raised-Cosines filters with different roll-offs:

As you can see, the cutoff frequency for all of these is always half the symbol rate, $\frac12 f_S = \frac12 \frac1{T}$, but the amount of energy outside of that depends on the roll-off. The closer you get to the perfect sinc shape in time ($\beta\rightarrow0$), the sharper that gets.

Notice that this leads to a (double-sided) bandwidth between the cutoff frequencies of $B=2\frac12f_S=f_S$!

However, even for the extreme case of a roll-off factor (which, by the way, is exactly defined to be the difference in occupied bandwidth compared to the minimal bandwidth given by above cutoff frequency) of $\beta=1$, you'd only get $B=2f_S$, so you actually consider the worst case for raised cosine filters – which is a justifiable simplification.

Notice that I'm presuming the book you're citing has a bandwidth definition that covers only a single side!

• Marcus- Can you clarify which "B" you are referring to? (Double-sided or single-sided and why you would use one over the other). The reason I ask is I typically think of the bandwidth of the PSK modulated signal as the double-sided bandwidth (when not specified otherwise). Or to describe simply if you are referring to "occupied bandwidth", the bandwidth the signal will occupy if at a carrier frequency. It seems you are referring to the single-sided bandwidth (or I misread) so this may create a point of confusion if not clarified. :) – Dan Boschen Apr 11 '17 at 11:58
• @DanBoschen ah, true, yes! I should actually use $B = 2f_S$, and mention that I consider bandwidth to be the full span from $-f_c$ to $+f_c$! – Marcus Müller Apr 12 '17 at 15:30
• @MarcusMüller I knew you knew! Just for the casual reader to not be off by a factor of 2 – Dan Boschen Apr 12 '17 at 16:51