FM Demodulation Using Phase Locked Loop
•
0 likes•90 views
This lecture discusses the demodulation of FM signals using phase locked loop, commonly known as PLL. Detailed mathematical modeling of the PLL is provided with some frequency domain approximations. The presentation is also available at youtube.com on following address: https://www.youtube.com/watch?v=CC0s_VCRwlI
More Related Content
Similar to FM Demodulation Using Phase Locked Loop
Similar to FM Demodulation Using Phase Locked Loop (20)
Recently uploaded
Recently uploaded (20)
FM Demodulation Using Phase Locked Loop
- 1. Phase Locked Loop FM Demodulation Also available: https://www.youtube.com/watch?v=CC0s_VCRwlI ْیِّلَصُن َو ُٗہدَمَْحنلَعیَکْال ہل ْوُسَراَّمَا ْمیر م ہاّٰللب ُذ ْوُعَاَف ُدْعَبَّالر نْطیَّشال َنمْسب ْمیج ْیحَّالر نمْحَّالر ہاّٰللم A Lecturer By Engineer Muhammad Abu Bakar Siddique
- 2. Phase Locked Loop (PLL) Phase Comparator Loop Filter 𝐺(s) Voltage Controlled Oscillator (VCO) sin 2𝜋𝑓𝑐 𝑡 + 𝜙 cos 2𝜋𝑓𝑐 𝑡 + 𝜙 𝑣
- 3. PLL is responsible for keeping heads on top of the screen and feet on bottom. Phase Locked Loop (PLL)
- 4. • Carrier recovery • Frequency (FM) demodulation • Frequency synthesis • Frequency amplifier • Clock recovery • Clock distribution Phase Locked Loop (PLL)
- 5. 𝜙 𝑒 = 𝜙 − 𝜙 𝑣 Phase Comparator aka Phase Detector Loop Filter 𝐺(s) Voltage Controlled Oscillator (VCO) 𝑣(𝑡) 𝑉 𝑠 = Φ 𝑒 𝑠 𝐺(𝑠) 𝑢 𝑡 = Acsin 2𝜋𝑓𝑐 𝑡 + 𝜙 𝜙 𝑣 𝑡 = 2𝜋𝑘 𝑣 𝑣 𝑡 𝑑𝑡 𝜙 𝑡 = 2𝜋𝑘 𝑓 𝑚 𝑡 𝑑𝑡 𝑦𝑣 𝑡 = Avcos 2𝜋𝑓𝑐 𝑡 + 𝜙 𝑣 FM Demod
- 6. PLL • FM modulated input: 𝑢 𝑡 = 𝐴 𝑐 sin 2𝜋𝑓𝑐 𝑡 + 2𝜋𝑘 𝑓∫ 𝑚 𝑡 𝑑𝑡 • Instantaneous frequency: 𝜔𝑖 = 𝑑 𝑑𝑡 (argument of sinusoid) • 𝜔𝑖 = 2𝜋𝑓𝑖 = 𝑑 𝑑𝑡 2𝜋𝑓𝑐 𝑡 + 2𝜋𝑘 𝑓∫ 𝑚 𝑡 𝑑𝑡 , ∵ 𝜔 = 2𝜋𝑓 • 2𝜋𝑓𝑖 = 2𝜋𝑓𝑐 + 2𝜋𝑘 𝑓 𝑚 𝑡 • Δ𝑓 ≡ 𝑓𝑖 − 𝑓𝑐 • 2𝜋𝑓𝑖 − 2𝜋𝑓𝑐 = 2𝜋𝑘 𝑓 𝑚 𝑡 • Δ𝑓 ∝ 𝑚(𝑡) for FM modulation
- 7. Voltage Controlled Oscillator (VCO) Voltage Controlled Oscillator (VCO) 𝑣(𝑡) 𝜙 𝑣 𝑡 = 2𝜋𝑘 𝑣 𝑣 𝑡 𝑑𝑡 𝑦𝑣 𝑡 = Avcos 2𝜋𝑓𝑐 𝑡 + 𝜙 𝑣 𝑓𝑣 = 𝑓𝑐 + 𝑘 𝑣 𝑣 𝑡
- 8. sin 𝜙 − 𝜙 𝑣 = sin 𝜙 − 𝜙 𝑣 + sin 4𝜋𝑓𝑐 𝑡 + 𝜙 + 𝜙 𝑣 Bandpass signal centered at 2𝑓𝑐 Lowpass signal centered at 0 ≈ 𝜙 − 𝜙 𝑣 = 𝜙 𝑒 Low Pass Filter Voltage Controlled Oscillator (VCO) 𝑢 𝑡 = Acsin 2𝜋𝑓𝑐 𝑡 + 𝜙 𝜙 𝑣 𝑡 = 2𝜋𝑘 𝑣 𝑣 𝑡 𝑑𝑡 𝜙 𝑡 = 2𝜋𝑘 𝑓 𝑚 𝑡 𝑑𝑡 𝑦𝑣 𝑡 = Avcos 2𝜋𝑓𝑐 𝑡 + 𝜙 𝑣 Phase Comparator Acsin 2𝜋𝑓𝑐 𝑡 + 𝜙 ⋅ Avcos 2𝜋𝑓𝑐 𝑡 + 𝜙 𝑣 ∵ cos 𝛼 sin 𝛽 = 1 2 sin 𝛼 + 𝛽 + 1 2 sin 𝛼 − 𝛽 lim 𝑥→0 sin 𝑥 𝑥 = 1 sin 𝑥 ≈ 𝑥
- 9. • 𝜙 𝑒 𝑡 = 𝜙 𝑡 − 𝜙 𝑣 𝑡 • 𝜙 𝑒 𝑡 = 𝜙 𝑡 − 2𝜋𝑘 𝑣∫ 𝑣(𝑡) 𝑑𝑡 • ∵ ∫ 𝑦 𝑡 𝑑𝑡 ↔ 𝑌 𝑠 𝑠 • ∴ Φ 𝑒 𝑠 = Φ 𝑠 − 2𝜋𝑘 𝑣 𝑉 𝑠 𝑠 • Φ 𝑒 𝑠 = Φ 𝑠 − 2𝜋𝑘 𝑣Φ 𝑒 𝑠 𝐺(𝑠) 𝑠 • Φ 𝑠 = Φ 𝑒 𝑠 1 + 2𝜋𝑘 𝑣 𝐺 𝑠 𝑠 • ∵ 2𝜋𝑘 𝑣 𝐺(𝑓) 𝑗2𝜋𝑓 ≫ 1, ∀ 𝑓 < 𝑊 • ∴ Φ 𝑠 ≈ Φ 𝑒 𝑠 2𝜋𝑘 𝑣 𝐺 𝑠 𝑠 FM demodulation 𝜙 𝑒 = 𝜙 − 𝜙 𝑣 Phase Comparator Loop Filter 𝐺(s) (VCO) 𝑣(𝑡) 𝑉 𝑠 = Φ 𝑒 𝑠 𝐺(𝑠)
- 10. • Φ 𝑠 = Φ 𝑒 𝑠 2𝜋𝑘 𝑣 𝐺 𝑠 𝑠 • Φ 𝑠 = Φ 𝑒 𝑠 𝐺 𝑠 2𝜋𝑘 𝑣 𝑠 • Φ 𝑠 = 𝑉 𝑠 2𝜋𝑘 𝑣 𝑠 • 𝑉 𝑠 = 1 2𝜋𝑘 𝑣 𝑠 Φ 𝑠 • ∵ 𝑑 𝑑𝑡 𝑦(𝑡) ↔ 𝑠𝑌(𝑠) • 𝑣 𝑡 = 1 2𝜋𝑘 𝑣 𝑑 𝑑𝑡 𝜙(𝑡) • ∵ 𝜙 𝑡 = 2𝜋𝑘 𝑓∫ 𝑚 𝑡 𝑑𝑡 • ∴ 𝑣 𝑡 = 𝑘 𝑓 𝑘 𝑣 𝑚 𝑡 • 𝑣 𝑡 ∝ 𝑚(𝑡) FM demodulation 𝜙 𝑒 = 𝜙 − 𝜙 𝑣 Phase Comparator Loop Filter 𝐺(s) (VCO) 𝑣(𝑡) 𝑉 𝑠 = Φ 𝑒 𝑠 𝐺(𝑠)
- 12. 𝜙 𝑒 = 𝜙 − 𝜙 𝑣 Phase Comparator Loop Filter 𝐺(s) (VCO) 𝑣(𝑡) 𝑉 𝑠 = Φ 𝑒 𝑠 𝐺(𝑠)
Editor's Notes
- نَحْمَدُہٗ وَ نُصَلِّیْ وَنُسَلِّمُ عَلٰی رَسُوْلِہِ الْکَرِیْمِ اَمَّا بَعْدُ فَاَعُوْذُ بِاللّٰہِ مِنَ الشَّیْطٰنِ الرَّجِیْمِ بِسْمِ اللّٰہِ الرَّحْمٰنِ الرَّحِیْمِ Dear viewers, this tutorial is about Phase Locked Loops and their role in demodulation of FM modulated signals.
- Phase locked loop is a circuit that synchronizes a locally generated signal to an input signal in frequency as well as in phase. In the synchronized, or the so-called locked state, the phase error between the local signal and the input signal is zero or very small. If a phase error builds up, the feedback control mechanisms redirects the local signal as to minimize the phase error with the input signal. The phase of the local signal is locked to the phase of the input signal hence the name phase-locked-loop. A PLL circuit consists of three basic components, a phase comparator, also known as phase detector, a loop filter and a voltage-controlled oscillator, more commonly know by its acronym VCO. We will discuss these components in more detail after a couple of slides. The phase detector compares the phase of the input signal against the phase of the locally generated signal of VCO. The output of the phase detector is proportional to the phase difference between the two inputs. The difference voltage is then cleaned by the loop filter. Output of the loop filter goes to the VCO input and the VCO frequency in the direction that reduces the phase difference between the input signal and the VCO output.
- PLLs were popularized by televisions and was responsible for keeping heads on top of the screen and feet on bottom. Otherwise there would have been a vertically scrolling picture. To enhance the quality of the TV receiver, the local oscillator responsible for vertical scan was phase-locked to the sync pulses, by so called “flywheel” synchronizers implemented by phase-locked loops. The term “flywheel” originates from the high inertia, of the synchronizer, enabling it to move through periods of high noise and weak input signal without loosing synchronization.
- Phase locked loops are used in television, cell phones, GPS, satellite receivers. PLL is used for carrier recovery, FM demodulation, frequency synthesis, generating higher frequencies from smaller stable crystal frequencies, recovery of clock signal from sequential digital data, clock distribution.
- To understand FM demodulation, assume an FM modulated signal u(t) is applied at input. The sinusoid carrier frequency is fc and its phase, phi(t), is proportional to the message signal m(t), integrated. The frequency sensitivity, 𝑘 𝑓 determines the depth of FM modulation. VCO output y-v(t) is another sinusoid of frequency fc and phase, phi-v(t) proportional to integral of VCO input v(t). VCO is just another FM modulator at the receiver. Phase comparator output is the phase error between the input FM signal u(t) and VCO output signal y-v(t). The loop filter has impulse response g(t). Its input is phase error 𝜙 𝑒 (t) and output is v(t). We will later see that under lock conditions, v(t) is proportional to m(t).
- The FM modulated signal u(t), a sinusoid with its argument given 2𝜋 𝑓 𝑐 𝑡+2𝜋 𝑘 𝑓 ∫𝑚 𝑡 𝑑𝑡 We define instantaneous frequency of a sinusoid as time derivative of its angle part 𝜔 𝑖 is the instantaneous frequency of incoming FM signal in radians per second, or fi Hertz, calculated as time derivative of argument of the sine function in u(t). We get 2𝜋𝑓 𝑖 =2𝜋 𝑓 𝑐 +2𝜋 𝑘 𝑓 𝑚 𝑡 We define the frequency deviation Δ𝑓 as difference between the carrier frequency fc and the instantaneous frequency fi, rearranging we get 2𝜋𝑓 𝑖 −2𝜋 𝑓 𝑐 =2𝜋 𝑘 𝑓 𝑚 𝑡 So, the frequency deviation is proportional to the message signal m(t). Frequency modulation is varying the frequency o the carrier signal around a center frequency proportional to the input message signal. Hence, we have correct mathematical representation of FM modulated m(t).
- Let us discuss the VCO. By definition, it is a circuit which has its frequency proportional to the applied input voltages v(t). In absence of v(t), VCO runs at its free running frequency, fc. 𝑘 𝑣 is the VCO deviation constant with units of Hz/V. For FM signals Its value is in range of 100,000 Hz/V. VCO output is again a sinusoid with phase angle proportional to the integral of VCO input v(t)
- The second component of PLL is phase comparator. It compares the phases of incoming FM signal u(t) and VCO output v(t) by employing a mixer, a device that mathematically multiplying the two signals. According to trigonometric product formula we obtain two sinusoids with sum and difference angles. The sum angle sinusoid has 4-pi-fc-t term, therefore its spectrum is centered at 2fc. A low pass filtered after mixer will stop this high frequency term, giving a sine function of phase difference. Under locked condition, this phase is small. Using the limit, limit x approaches zero sin x over x equals 1. The phase comparator output is proportional to the phase difference, from now onwards known as phase error.
- Now let us see how the FM demodulation takes place. So phi-e-t is the phase difference between the phases of FM input and VCO output Which after substituting the phi-v-t with its value, the integrated VCO input v(t) and taking the Laplace transform we get Where we use the fact that integration is replaced by a pole at zero in Laplace transform Noting that the VCO input V(s) is the product of loop filter input and its impulse response in frequency domain, V(s) is replace with phi-e-s times G-s We get the phi-s as function of phi-e-s Since Fourier transform is Laplace transform with s replaced by j-2-pi-f and noting that VCO deviation constant kv is a large number with value of G(S) close to unity in pass band. The fraction term is much greater than unity within the signal bandwidth. Therefore Phi-s can be approximated as
- No again using the fact that product phi-e-s and G-s gives the VCO input V(S), Rearranging to write V(s) as function of phi-s And noting the derivative in time domain is multiplication by s in Laplace domain, The inverse Laplace transform gives v(t ) proportional to the derivative of phi(t) Since phi-t originally was phase of the incoming FM modulated carrier, and that phase was proportional to the integral of the modulating signal m(t) We obtain v(t) which is given in terms of m(t). K_f and K_v both are constants Therefore v(t) is proportional to m(t), which is our desired demodulated signal.