# Transmission Bandwidth and Sampling frequency

In LTE standards, for a transmission bandwith of 2.5 MHz, the sampling frequency is 3.84 MHz. I was wondering with this choice, are we violating Nyquist theorem?

I found the details in the following link (page 15)

http://www.freescale.com/files/wireless_comm/doc/white_paper/3GPPEVOLUTIONWP.pdf

I would have assumed that with a bandwith of 2.5 MHz, then the channel sampling frequency (signal input sampling frequency) should be at least 5 MHz?

Thanks.

• With complex/IQ sampling at 3.84 MHz, one is really sampling at 7.68M real values per second (one of them labeled imaginary). – hotpaw2 Apr 20 '15 at 21:08

Complex (I/Q) signal processing is used in this case. With a complex-valued signal sampled at a rate $B \text{ Hz}$, you can unambiguously represent a total bandwidth of $B \text{ Hz}$.

What you're thinking of is the real-valued signal case, where a signal sampled at $B \text{ Hz}$ can unambiguously represent a total bandwidth of $\frac{B}{2} \text{ Hz}$. This is the most common statement of the Nyquist theorem, but it only holds for real-valued signals.

Note that the total number of "values" per second is the same, as each complex sample consists of a real and imaginary part. So, there's no "information gain" to be made by using complex sampling, it's just more convenient for many applications (including digital communications signal processing).

• Interesting, so Nyquist rate that I mentioned is defined for real valued signals, for complex valued signals what happens to Nyquist rate? So the specific example that I gave above means that we need to sample the transmitted signal at 3.84 MHz and this would guarantee no aliasing for a channel transmission bandwidth of 3.84 MHz? Is my understanding correct? @Jason R – Tyrone Apr 20 '15 at 20:55
• That's correct. – Jason R Apr 20 '15 at 21:07

I have a bit different take on the whole sampling thing. It goes like this. If you observe carefully LTE 20MHz is not one single carrier modulated with 20 MHz (18MHz actually) of spreaded bandwidth. If I just consider one subcarrier out of 2048, the information bandwidth is 15kHz only. This 15kHz bandwidth is same for 1st (at-9 MHz) middle (0) or last (+9MHz) subcarrier.

Now if I have a 15kHz carrier at baseband all I need is sampling of 30kHz for real signal or 15kHz for complex signal.

We shall assume that max signal which we want to faithfully sample is of 15kHz Bandwidth but at 18 MHz carrier.

To sample that I need at least 18MHz sample rate as signal is complex. There may be many undersampling rate (in different nyquist zone) possible as the bandwidth is too small compared to carrier but sampling rate must be chosen so that it satisfy sampling criterion for all the sub carrier.

Hence something more than 18MHz shall satisfy Nyqusit (first zone) for last subcarrier as well as all subcarriers.

Now to answer the question of why 30.72 and not 18 or 20Mhz sampling rate, we shall understand the system evaluation. LTE is evaluation from WCDMA, where base rate is 3.84MCps (Mega Chips per second). All the circuitory and clocks are ready build and designed with this rate. So to chose a rate more than 18 MHz we shall use a good multultiple of 3.84, which if you see is 8 here (30.72/3.84). And this is how everything converge to this number.

Comments are welcome on my take.