Why a control loop and not a filter? We have a narrow bandwidth signal with an unknown frequency f between f1 and f2. Motivation consider this scenario- Phase - Locked loops PLL PLL structure Frequency-Locked Loop FLL Frequency modulation - FM "conveys information over a carrier wave by varying its instantaneous frequency."

wikipedia Lior Waldman Seminar DSP 2013 Stewie sends Brian a digital information in a constant rate of 1 bit every T seconds. Stewie's transmitter has a clock that sends a bit every T seconds. Brian knows that Stewie is sending him a bit every T seconds However the clock in Brian's receiver is independent and runs in a slightly different rate so instead of looking for a bit every T seconds it looks every T ' After enough time had passed the receiver is looking for a bit in the wrong time we have a “bit slips” - we will either miss bits or report incorrect ones. What can Stewie and Brian do? Stewie can send a second signal (a sinusoid, Rectangular, pulse) of frequency f generated by the transmitter's clock. the receiver needs to set its clock according to this sinusoid and the discrepancy problem vanishes. V (t) e(t) V (t) Voltage-Controlled

Oscillator in out Phase

Detector Loop Filter control

signal The Phase Detector This block has 2 inputs: the input reference signal whom we want to treck.

the output of the V.C.O V (t) in V (t) out The block's output: the phase error, the difference between the 2 input signals. e(t) e(t) V (t) in V (t) out Implementation of the Phase Detector block xor gate: V (t) in V (t) out e(t) e(t)= { 1 if V (t) out V (t) out = 0 otherwise The loop filter Input:

- the output of the Phase detector e(t) Purpose:

a low-pass filter needed to remove/reduce background noise-

remove jitter and pass wander

outputs voltage representing the difference between the two signals to pass to the V.C.O The Voltage - Controlled Oscillator

V.C.O Input:

Control signal from the loop filter representing the difference between the two signals.

Output:

the receiver's generated signal. V (t) e(t) V (t) Voltage-Controlled

Oscillator in out Frequency

Detector Integrator control

signal The Frequency Detector Voltage-Controlled Oscillator (VCO). Output:

a signal to match the input How does it work?

if the input is zero the VCO oscillates at its natural frequency

if the input is nonzero the outputs frequency changes to W (t) 0 x(t) W + V(t) 0 Integrator if the input frequency is

the output of the frequency diff block and the input to the VCO is

the VCO changes its frequency to W > Wo W-Wo Wo+W-Wo = W this is a momentary change and return to the natural frequency. that why we need the integrator maintains a constant value when the difference becomes zero, forcing the VCO to remain at W Is this enough for Stewie and Brian? Now we are locked to the frequency but the phase of the input and output is not the same Brian will not know when exactly the bits arrive Another implementation Analog Multiplier: e(t)=Vin(t)*Vout(t) =sin(2*pi*f*t + phi1)

=cos(2*pi*f*t + phi2) Vin(t)*Vout(t)= Vin Vout sin(phi1-phi2)/2 + sin(4*pi*f*t +phi1+phi2)/2 Using Low pass Filter, we can block the high frequency term. Then, for small phase difference, we have the output as = sin(phi1-phi2) e(t) PLL is not linear PLL is a control system that generates an output signal whose phase is related to the phase of an input "reference" signal. ~ (phi1-phi2) i have a data signal i want to send.- X (t)

and a carrier signal X (t) = Acos(2*pi*f *t)

i want to combine them to one signal- c c m Example usage: IN simple words: as you can see the information i want to send is hidden in the signal's phase, i will use PLL in a different way then before, i will use it to extract the phase! Let the output from VCO is given as y(t) = cos(2*pi*f *t + phi2) c phi1 we will use the analog implementation of the phase detector e(t)=x(t)*y(t) X(t) and after the loop filter we remain with = sin(phi1-phi2) e(t) In general, the Loop Filter output will tend to follow the derivative of the input phase function control signal = (phi1)' / 2*pi*f Loop Filter output = demodulated FM Signal! f is the frequency deviation, The differences between Analog and Digital PLL Input: Digital- a square signal Analog- a sinusoid signal Digital signal XOR implementation: not a filter FLL are used in radio, telecommunications and other electronic applications to generate stable frequencies, or to recover a signal from a noisy communication channel. role: measures the relationship between the reference input signal and the output of the oscillator.

the frequency detector measures the difference of frequency at its 2 inputs and generates an output that represents the ratio of this 2 frequencies. The input signal can be a sinusoid, rectangular or pulse signal The Frequency Detector have different implementations: Can implement a phase detector and use its derivative

Sin- Two frequency demodulators and an adder with one input negated. This two implementations causes the PLL to be non linear. The bast a band pass filter can do is filter between f1 to f2 and pass the signal including the noise in that range A control loop (like PLL) can search between f1 and f2 and open a very narrow "window" around f and continue following it. control loops- filter- Lecture's Outline Explanation: Let v = (s1,s2) vector of two signals and F is a linear function then F(av) = aF(v) Let G be the analog multiplier function G(av) = G(as1,as2) = a s1*s2 = a G(v) 2 2 Modulation- Modulation is the process of conveying a message signal, for example a digital bit stream or an analog audio signal, inside another signal that can be physically transmitted Analog modulation methods- Amplitude modulation

Phase modulation frequency modulation- =cos(2*pi*f*t + phi2) Vout If analog- phi2 = 2*pi*f u + phi2_old loop filter output = control signal = u(t) Questions? Thank you Control loops vs filters

Motivation

FLL

PLL

Usage of PLL

GPS Lock

FM

FM Demo Pulse-

We negate one of the signals

Construct integral

Construct another integral on the results. this represent the phase change

The phase derivative is the frequency change. For large differences we can use arcsin Noise comes from temperature changes, imperfections in materials, aging, and external influences suffered by the signal before it reaches the PLL input. The variation of the transmitted signal over time is conventionally divided into two components: Wander expresses slow, smooth drifting of signal rate. Jitter conveys fast, erratic jumps The border between the two components is conventionally set at 10 Hz. If we reached a situation where there is only a small frequency change the FLL becomes an open loop with no feed-back and cannot correct any errors that it could make If there is a small frequency offset eventually it will increase to a phase difference and will be fixed

If we reached a situation where there is only a small phase difference it will not be fixed and the feed back process remains.

That is why PLL stays a closed loop system GPS LOCK Global Positioning System (GPS) Developed in 1973 by the U.S. Department of Defense.

It became fully operational in 1994

originally run with 24 satellites (Currently in orbit- 31)

Orbiting at an altitude of approximately 20,200 km each SV makes two complete orbits each sidereal day. GPS is not the only navigation system.

GLONASS- Russian, 18 satellites

GLILEO- European, 4 satellites

Beidou- Chinese, 10 satellites, offering services to customers in the Asia-Pacific region How does it work? Each satellite sends a signal with its exact location and the time the message was transmitted.

A receiver picks up the signal.

And using the time that passed and location calculate the distance to the satellite.

We need 4 satellites to calculate the receiver's exact location. What does it sends? Carrier wave frequency is 1575MHz

Two digital series of (-1,1) at low frequency

Data- location and time at 50Hz

1023 bit long pseudo-random number (PRN) sequence sent at a rate of 1.023 megabits/sec How a GPS Receiver Gets a Lock? After the receiver "smells" a signal:

The signal's carrier and code frequencies change due to the Doppler effect, which is caused by the motion of

the GPS satellite as well as from the motion of the GPS receiver.

In order to track the GPS signal, the C/A code information must be removed. As a result, it requires two phase-locked loops to track a GPS signal. One is a phase-locked loop in the frequency domain and the other is a time-locked loop in the 1023 bit code space domain

with the goal of tracking both the code and carrier phases for that signal. If digital- NCO- numerically controlled oscillator NCO Cont. Input-

N- the number of bits carried in the phase accumulator. N sets the NCO frequency resolution

frequency control word - size to add Each clock cycle produces a new N-bit output consisting of the previous output obtained from the register summed with the frequency control word (FCW) which is constant for a given output frequency Then we use this N bits to extract the generated signal value from a look up table

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