# Matlab Phase Locked Loop Design : FM Demodulation

EDIT: It seems I've asked too much at once here. I'll do some more studying and come back if I have more specific questions.

I'm trying to design an analog phase locked loop in Matlab. I've read around and there are a few examples of this out there but I'm not sure how a few things work or how to adapt it for my use. I'll be asking a lot of questions so feel free to answer one or all.

This is a block diagram of the design I am constrained to:

Where my input signal X(t) is modulated using Matlab's built in FMMOD and the original audio has a sampling frequency of 44.1kHz.

I have questions about each block, and about PLL demodulation in general. I will first list constraints about my system:

• The peak frequency deviation must be 75kHz

• The loop filter transfer function must be: $$\frac{s+a}{s}$$

• The VCO must be used as feedback. I say this as some designs have the output of the VCO be the demodulated message.

The questions I have about the phase detector are:

• The phase detector consists of a multiplier whose output is fed into a lowpass filter. The purpose of this is to isolate the difference between the incoming signal and the estimated signal. I understand the math behind why a multiplier is used, since it will have two components: 1 of double the FM carrier frequency and 1 of the difference between the incoming message and the estimated message with the carrier frequency eliminated. Is this understanding correct? For example, if I multiplied a modulated 100Hz and 500Hz sinusoid, the output of the phase detector would be a constant value?

• How would one determine the cutoff frequency of this lowpass filter? Should it be the maximum frequency of the demodulated message?

• Later in the system design, I believe it will be important to know the gain o the phase detector. How is this determined?

• It is my understanding that in this design, the loop filter must be designed so that the transfer function of $$\frac{M(f)}{X(f)}$$ sees a slope of 20 dB /decade. (I will attach an image of my math to find this transfer function at the end of this post). This is because during the process of FM modulation, the original message is integrated put in the phase of the carrier signal. This means that to undo this integration, its derivative will need to be taken. Is this understanding correct?

• I understand the VCO's role in this the least. I believe it should be nearly identical to an FM modulator. This means it is FM modulating the estimated message and feeding that modulated signal to the phase detector. Is this so that the phase detector can determine the current difference between the incoming message and the estimated message?

• Since the VCO is essentially an FM modulator, it will also need a peak deviation frequency. If I have designed the system correctly, the input into the VCO should be the derivative of the FM modulated signal (which is the original message). Meaning its peak phase deviation should be identical to the FM modulation used for the actual message. In this case it should be 75kHz. Is this understanding correct?

• Can the FMMOD Matlab function be used to simulate the VCO in this case? Can I simply input 75KHz into the peak frequency deviation argument of this function?

• It is my understanding that the VCO will produce some gain I will call Kvco during the modulated process. How is this calculated? Or if it is not realistically worth it to calculate, how is this gain accounted for?

Questions about the Amplifier gain = u:

• I am not sure if I am calculating the gain of this system correctly, or understanding how it comes into play. The carrier signal has a gain Ac, the phase detector has some gain Kd, the carrier signal of the VCO output is Av, the gain of the VCO modulation itself is Kvco, the gain of the amplifier is Au. There will also be a gain of 0.5 from the trigonometry used in the phase detector. This means that the gain of the system is: $$\frac{Kd * Kvco * Ac * Av * Au}{2}$$

• How are the gain elements Ac, Av, and Au (and Kvco if it is controllable) selected? I know that the larger the gain the faster the loop begins tracking the original message, but is there a limit to this? Does this gain effect the actual amplitude of the estimated message m(t)? How should I design the gain of this system?

• Will I need to normalize the amplitude of the estimated message in some way to account for this system gain? I am confused on if the gain I described above only effects the loop tracking or if it also effects the message amplitude.

MATH TO DETERMINE TRANSFER FUNCTION TO SOLVE FOR 'a'

• Hints - once loop bandwidth is specified, use that to determine loop gain, which determines BW in conjunction with VCO's falling gain. Forget loop filter for the moment, it complicates things and is not needed for a) operation or b) stability. When you add it it's for c) performance. The loop filter needs unity gain at loop BW anyway, so does not alter loop BW. Define PSD gain in the normal way - make an input change, measure the output change, gain is their ratio! – Neil_UK Jun 2 '18 at 5:21
• Far too many questions. VTC as too broad. Keep simple and respect the fact that this is a question and answer site and not an on-line training facility. – Andy aka Jun 2 '18 at 10:25
• Too many questions, it would be easier for contributors if you ask single question at one time – Avionics Engg Sep 3 '18 at 13:54