# Constant Envelope Modulation Techniques for Limited Power Millimeter Wave Links
## 3 Error Rate Degradation in the Presence of Co-Channel Interference  
Co-channel interference occurs between two transmitters that are using the same frequency. The co-channel interference can severely affect the performance of the symbol error rate (SER). In this section, the SER performance properties of the CE-MTO and the CE-OFDM in the presence of co-channel interference are studied. The signal arriving at the receiver from the transmitter is a constant envelope phase-modulated waveform. Its complex amplitude at baseband is given by the following [25]:  
\[\mathrm{s}(t)=A_{c}e^{j\phi(t)} \tag{4}\]  
where \(j=\sqrt{-1}\) and \(A_{c}\) is the (constant) signal envelope. \(\phi(t)\) is the information-bearing message phase. Assuming an in-band interference, its respective complex amplitude is as follows:  
\[\iota(t)=\frac{A_{c}}{\sqrt{SIR}}e^{j\theta(t)} \tag{5}\]  
with the power given by the variance, as follows:  
\[E\big{[}t^{2}(t)\big{]}=\frac{A_{c}^{2}}{SIR} \tag{6}\]  
where we define SIR as the signal to interference power ratio at the receiver site, as in Equation (3). The interfering signal phase, \(\theta(t)\), is a stochastic process that is uniformly distributed \((-\pi,\pi)\). Both the signal and interference are received simultaneously, resulting in a composite waveform, written as follows:  
\[\mathrm{r}\big{(}t\big{)}=\mathrm{s}(t)+\iota(t)=A_{c}\cdot\bigg{[}e^{\mu(t)}+ \frac{1}{\sqrt{SIR}}e^{\mu(t)}\bigg{]}=\underbrace{A_{c}\bigg{[}\cos\big{[} \phi(t)\big{]}+\frac{1}{\sqrt{SIR}}\cos\big{[}\theta(t)\big{]}\bigg{]}}_{\text{ \begin{subarray}{c}i(t)\end{subarray} }}+\underbrace{J_{c}\big{[}\sin\big{[} \phi(t)\big{]}+\frac{1}{\sqrt{SIR}}\sin\big{[}\theta(t)\big{]}\bigg{]}}_{ \text{ \begin{subarray}{c}i(t)\end{subarray} }} \tag{7}\]  
Figure 2: Illustration of the signal transmission in the presence of interference. EIRP—effective isotropic radiated power. where its phase is as follows:  
\[\hat{\phi}(t)=\arctan\left[\frac{Q(t)}{I(t)}\right]=\arctan\left\{\frac{\sin[ \phi(t)]+\frac{1}{\sqrt{SIR}}\sin[\theta(t)]}{\cos[\phi(t)]+\frac{1}{\sqrt{SIR}} \cos[\theta(t)]}\right\}. \tag{8}\]  
Note that when \(SIR\rightarrow\infty\), the resulted phase is \(\hat{\phi}(t)=\phi(t)\), equal to that of the required signal, as expected. In order to calculate the interference component in the phase, we assume \(\phi(t)=0\). In this case, the phase fluctuation due to the interferer is as follows:  
\[\xi(t)=\arctan\left\{\frac{\sin[\theta(t)]}{\sqrt{SIR}+\cos[\theta(t)]}\right\}. \tag{9}\]  
Further analysis considers the detection of orthogonal modulation techniques (CE-OFDM [25] and CE-MTO [26] techniques). In the receiver (see Figure 3), after an analog to digital conversion, the samples, \(r[n]\), are sent to the phase demodulator. The output of the phase demodulator in the continuous-time presentation is \(\hat{\phi}(t)=\phi(t)+\xi(t)\), where \(\xi(t)\) is the phase interference component given by Equation (9). The amplitude is kept constant by a limiter. A set of matched filters calculate the correlations, as follows:  
\[\mathbf{Q}[k]=\sum\limits_{n=1}^{N_{chip}}\hat{\phi}[n]q_{k}[n]=\sum\limits_{n=1}^ {N_{chip}}\left\{\phi[n]+\xi[n]\right\}q_{k}[n]=\mathbf{S}[k]+\mathrm{I}[k] \tag{10}\]  
where \(\phi[n]=2\pi hC\left\{\sum\limits_{k=1}^{N}I_{k}\cdot q_{k}[n]\right\}\), \(h\) is the modulation index, and \(C\) is a constant used to normalize the variance of the resulted phase, namely: \(\{I_{k}\}_{k=1}^{N}\in\{\pm 1,\pm 3,\ldots,\pm(M-1)\}\), where M is the pulse-amplitude modulation (PAM) constellation, \(I_{k}\) is the \(N\) real valued data symbols, and \(q_{k}[n]\) is the discrete orthogonal waveforms. In the CE-OFDM technique, \(q_{k}[n]\) is the orthogonal subcarriers' quadrature components, while in CE-MTO \(q_{k}[n]\) is the orthogonal series generated by the Hadamard matrix.