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# Non coherent demodulation of pi/4 DQPSK (TETRA)

In TETRA specifications, one of the modulation technique used is $\frac{\pi}{4}$ Differential Quaternary Phase Shift Keying (DQPSK). We will discuss the bit error rate with non-coherent demodulation of $\frac{\pi}{4}$DQPSK in Additive White Gaussian Noise (AWGN) channel.

## pi/4 DQPSK

The incoming bit sequence is grouped into two bits and is mapped into differential phase as follows:

 B(2k-1) B(2k) $D\phi(k)$ 1 1 $-\frac{3\pi}{4}$ 0 1 $+\frac{3\pi}{4}$ 0 0 $+\frac{\pi}{4}$ 1 0 $-\frac{\pi}{4}$

Table : Phase transitions for $\frac{\pi}{4}$ DQPSK modulation (Ref Table 5.1 of ETSI 301-893 V3.2.1)

The modulation symbol $S(k)$ is formed by applying a phase offset to previous symbol $S(k-1)$ and is defined as follows:

$S(k) = S(k-1) e^{jD\phi(k)}$ and

$S(0) = 1$.

Alternately, the phase transitions can be represented as

$\phi(k) = \phi(k-1) + D\phi(k)$.

The constellation of the $\frac{\pi}{4}$ DQPSK occupies phase values separated by $\frac{\pi}{4}$as shown below in the blue dots. The red lines shows all possible phase transitions.

Figure: Constellation of $\frac{\pi}{4}$ DQPSK (Ref Figure 5.1 of ETSI 301-893 V3.2.1)

## Channel Model

The transmitted waveform gets corrupted by noise $n$, typically referred to as Additive White Gaussian Noise (AWGN).

Additive : As the noise gets ‘added’ (and not multiplied) to the received signal

White : The spectrum of the noise if flat for all frequencies.

Gaussian : The values of the noise $n$ follows the Gaussian probability distribution function, $p(n) = \frac{1}{\sqrt{2\pi\sigma^2}}e^{\frac{-(n-\mu)^2}{2\sigma^2}$

with mean $\mu=0$ and

variance $\sigma^2 = \frac{N_0}{2}$.

The received symbol is,

$y(k) = S(k) + n(k)$

## Non Coherent Receiver

A non-coherent receiver relies on the phase transitions between consecutive symbols to form an estimate of the transmitted bits. The sequence of operation is as follows:

a) On the received symbols estimate the phase

$\hat{\phi}(k) = \angle y(k)$

b) Find the delta of the estimated phase between consecutive symbols

$\hat{D}\phi(k) = \hat{\phi}(k)-\hat{\phi}(k-1)$

c) Quantize the estimated delta phase values lying from $[-\pi,\mbox{ } +\pi)$ as follows and convert to bits per the following encoding:

$\begin{array}
-\pi & \le & \hat{D}\phi(k) & < & -\frac{\pi}{2} & \right & -\frac{3\pi}{4}& : & 11\\
-\frac{\pi}{2} & \le & \hat{D}\phi(k) & < & 0 & \right & -\frac{\pi}{4}& : & 10\\
0 & \le &\hat{D}\phi(k) & < & \frac{\pi}{2} & \right & +\frac{\pi}{4}& : & 00\\ \frac{\pi}{2} & \le & \hat{D}\phi(k) & <& \pi & \right & +\frac{3\pi}{4}& : &01\\ \end{array}$
.

Figure: Received delta phase to bit mapping

The theoretical bit error rate defined with the mapping above is (Ref Chapter 5.2.8 of Digital Communications by Proakis, 4th Edition)
$P_{non-coh,dqpsk} = \mathbf{Q}_{1}(a,b) - \frac{1}{2}\mathbf{I}_0(ab)e^{-\frac{(a^2+b^2)}{2}}$,

where

$a = \sqrt{\frac{2E_b}{N_0}\left(1-\frac{1}{\sqrt{2}}\right)}$ ,

$b = \sqrt{\frac{2E_b}{N_0}\left(1+\frac{1}{\sqrt{2}}\right)}$ ,

$\mathbf{Q}_{1}(a,b) = e^{-\frac{(a^2+b^2)}{2}}\sum_{k=0}^{\infty}\left(\frac{a}{b}\right)^k\mathbf{I}_k(ab)$ and

$\mathbf{I}_k(x)$ is the modified Bessel function of first kind.

## Simulation results

The script performs the following

(a) Generate random binary sequence of +1’s and 0’s.

(b) Group two bits together and apply $\frac{\pi}{4}$ DQPSK modulation

(c) Add white Gaussian noise.

(d) At the receiver, estimate the phase of the received symbols. Based on the delta of the received phase, estimate the transmitted bits

(e) Repeat for multiple values of $\frac{E_b}{N_0}$ and plot the simulation and theoretical results.

Click here to download the script for simulating BER of pi/4 DQPSK with non-coherent demodulation at the receiver

Figure: BER plot for $\frac{\pi}{4}$ DQPSK with non-coherent demodulation

## Observations

Can see good agreement between simulated and theoretical results. ðŸ™‚

## Reference

Digital Communications by Proakis, 4th Edition

## 9 thoughts on “Non coherent demodulation of pi/4 DQPSK (TETRA)”

1. bai says:

Hi, what is the different between pi/4 DQPSK and DQPSK ? How can i change the code to make it work for DQPSK ? Thanks

2. Bai says:

Hi, may I know how is it different from DQPSK, how can I change the code to get the BER for DQPSK ? Please advise.

3. dule markovic, MSEE says:

Thank you very much for this article. It’s very nice

Dule Markovic, Belgrade, Serbia

4. Ivy says:

Hi Krishna,
Your blog helps me a lot!
Thanks a lot!
Now I’m still cofused by the modulation and demodualtion about D8PSK, do you have some articles and matlab codes about it?
Hoping for your help!
Thanks!

1. @Ivy: Well, I have not discussed D8PSK till date. Quite likely in the near future.

5. Ankur says:

Hi Krishna,

Thanks for sharing some concepts on pi/4 DQPSK. The article was really helpful. I would like to know if you would be doing any codes or if you can share some concepts on star QAM and PSAM (Pilot Symbol Assisted modulation).

1. @Ankur: Can you please point me to literature describing star QAM and PSAM