- [[#Sine-wave Subcarrier]] - [[#Getting the Bessel Function]] - [[#Conversion to Power Components]] - [[#Pt Calculation]] - [[#Pc Calculation]] - [[#Pd Calculation]] - [[#Square-wave Subcarrier:]] The general equation for phase modulation is: $s(t)=A\cos(2\pi f_ct+k_pm(t))$ Where: - $s(t)$ = the phase modulated signal - $A$ = amplitude of the carrier - $f_c$ = carrier frequency - $k_p$ = phase sensitivity in rads per unit of $m(t)$ - $m(t)$ = modulating signal The modulating signal can either be a subcarrier with modulation or modulation directly on the carrier: 1. Data modulated onto a 1. Sine-wave subcarrier 2. Square-wave subcarrier 2. No subcarrier - direct on carrier The square-wave subcarrier actually reduces down to the no subcarrier variant - will show that later. ## Sine-wave Subcarrier ### Getting the Bessel Function $m(t)=M\sin(2\pi f_mt)$ Where: - $M$ is the amplitude of the subcarrier - $f_m$ is the frequency of the subcarrier The equation for the phase modulation is now: $s(t)=A\cos(2\pi f_ct+k_pM\sin(2\pi f_mt))$ We can define the modulation index as: $\beta=k_pM$ Which simplifies the equation to: $s(t)=A\cos(2\pi f_ct+\beta\sin(2\pi f_mt))$ Use Euler’s Formula: $⁍$ $s(t)=A\frac{e^{i(2\pi f_ct+\beta\sin(2\pi f_mt))}+e^{-i(2\pi f_ct+\beta\sin(2\pi f_mt))}}{2}$ $s(t)=\frac{A}{2}(e^{i2\pi f_ct}e^{i\beta\sin(2\pi f_mt)}+e^{-i2\pi f_ct}e^{-i\beta\sin(2\pi f_mt)})$ We now invoke the **Jacobi-Anger expansion**: $e^{iz\sin \theta}=\sum_{n=-\infin}^{\infin}J_n(z)e^{in\theta}$ Which we can see that: $e^{i\beta\sin(2\pi f_mt)}=\sum_{n=-\infin}^{\infin}J_n(\beta)e^{in2\pi f_mt}$ Putting that back into the formula and pulling out the like terms: $s(t)=\frac{A}{2}\sum_{n=-\infin}^{\infin}J_n(\beta)(e^{i2\pi f_ct}e^{in2\pi f_mt}+e^{-i2\pi f_ct}e^{-in2\pi f_mt})$ Bringing back Euler’s Identity, it simplifies to: $s(t)=A\sum_{n=-\infin}^{\infin}J_n(\beta)\cos(2\pi t( f_c+nf_m))$ How do we interpret this? The carrier is now a sum of cosine waves with amplitude $\frac{AJ_n(\beta)}{2}$and frequency $f_c+nf_m$. ### Conversion to Power Components For a standard cosine wave: $⁍$ The average power is: $P_n=\frac{A_n^2}{2}$ For our equation with the sine-wave subcarrier, the power is: $P_n=\frac{A^2J_n^2(\beta)}{2}$ ### Pt Calculation The total power is the sum of the powers of each part: $P_T=\frac{A^2}{2}\sum_{n=-\infin}^{\infin}J_n^2(\beta)$ The Bessel Identity equals 1 and therefore: $P_T=\frac{A^2}{2}$ We can get the generic formula $\frac{P_n}{P_T}=\frac{\frac{A^2}{2}J_n^2(\beta)}{\frac{A^2}{2}}=J_n^2(\beta)$ ### Pc Calculation The power in the carrier is such at that the only frequency is that of the carrier. We see that the only $n$ such that $f_c+nf_m=f_c$ is where $n=0$ and therefore: $\frac{P_C}{P_T}=\frac{P_0}{P_T}=J_0^2(\beta)$ ### Pd Calculation The power in the data is just any power that is not in the carrier. $\frac{P_D}{P_T}=1-J_0^2(\beta)$ _However_ according to the Blue Books, for DSN only the first sidelobe is used: $\frac{P_D}{P_T}=J_1^2(\beta)$ ## Square-wave Subcarrier: $m(t)= \begin{cases} M, & \text{for } La < t < L(a+1) \\ -M, & \text{otherwise}\end{cases}$ Where: - $L$ = half period of the square-wave - $a$ = an even integer When can get $\beta$ again and substitute back into the original equation $s(t)=A\cos(2\pi f_ct\pm \beta)$ Recall the average power of a cosine wave, which is not dependent on the phase. Therefore $P_T$ is the same as a simple sine-wave. We see that regardless of $\beta$, the frequency is always that of the carrier and thus all the power is in the carrier (but how can we get power in the data lol?) ![[Connect#Connect]]