Analog Communication (EC-403) - Unit 4 Short Notes

UNIT 4: ANALOG COMMUNICATION - EXAM-FOCUSED SHORT NOTES

1.0 SIGNAL ANALYSIS & FOURIER TRANSFORM

1.1 Fourier Transform (FT) Fundamentals

• Definition: The Fourier Transform (FT) of a continuous-time signal \( x(t) \) is its frequency-domain representation:

$$X(f) = \int_{-\infty}^{\infty} x(t) e^{-j2\pi ft} dt$$

It decomposes a signal into its constituent frequencies, crucial for analyzing modulation spectra and system responses.

• Key Properties (with Time-Domain ↔ Frequency-Domain mapping):

  | Property | Time-Domain | Frequency-Domain | | :--- | :--- | :--- | | Linearity | \(a x_1(t) + b x_2(t)\) | \(a X_1(f) + b X_2(f)\) | | Time Shifting | \(x(t - t_0)\) | \(X(f) e^{-j2\pi f t_0}\) | | Frequency Shifting | \(x(t) e^{j2\pi f_0 t}\) | \(X(f - f_0)\) | | Time Scaling | \(x(at)\) | \(\frac{1}{|a|} X\left(\frac{f}{a}\right)\) | | Duality | If \(x(t) \leftrightarrow X(f)\) | Then \(X(t) \leftrightarrow x(-f)\) | | Convolution | \(x_1(t) * x_2(t)\) | \(X_1(f) X_2(f)\) | | Parseval's Theorem | \(\int |x(t)|^2 dt\) | \(\int |X(f)|^2 df\) (Energy) |

• Exhaustive FT Pairs (Must Memorize):

  | Signal \(x(t)\) | Fourier Transform \(X(f)\) | | :--- | :--- | | Unit Impulse \(\delta(t)\) | \(1\) | | Unit Step \(u(t)\) | \(\frac{1}{j2\pi f} + \frac{1}{2}\delta(f)\) | | Signum \(\text{sgn}(t)\) | \(\frac{1}{j\pi f}\) | | Sinusoid \(\cos(2\pi f_0 t)\) | \(\frac{1}{2}[\delta(f-f_0) + \delta(f+f_0)]\) | | Sinusoid \(\sin(2\pi f_0 t)\) | \(\frac{1}{j2}[\delta(f-f_0) - \delta(f+f_0)]\) | | Gate/Rect \(\text{rect}\left(\frac{t}{\tau}\right)\) | \(\tau \text{sinc}(f\tau)\) | | Damped Sinusoid \(e^{-at}u(t)\) | \(\frac{1}{a + j2\pi f}\) (for \(a>0\)) |

[!TIP] Exam Focus: Questions on FT of gate function, damped sinusoid, and properties (especially time/frequency shifting and duality) are very frequent. Practice proving properties.

1.2 Correlation Functions & Spectral Densities

• Auto-correlation \(R_{xx}(\tau)\): Measure of similarity of a signal with a time-shifted version of itself.

$$R_{xx}(\tau) = \int_{-\infty}^{\infty} x(t) x^*(t-\tau) dt \quad \text{(Energy)}$$

$$R_{xx}(\tau) = \lim_{T\to\infty} \frac{1}{T} \int_{-T/2}^{T/2} x(t) x^*(t-\tau) dt \quad \text{(Power)}$$

*   **Properties:** Even function, maximum at \(\tau=0\), \(R_{xx}(0)\) = signal energy/power.

*   **Wiener-Khinchin Theorem:** The Fourier Transform of the auto-correlation function is the **Energy/Power Spectral Density (ESD/PSD)**.

$$\mathcal{F}\{R_{xx}(\tau)\} = S_{xx}(f) = |X(f)|^2 \quad \text{(ESD)}$$

    This links time-domain correlation to frequency-domain power distribution.

• Cross-correlation \(R_{xy}(\tau)\): Similarity between two different signals.

$$R_{xy}(\tau) = \int_{-\infty}^{\infty} x(t) y^*(t-\tau) dt$$

*   **Property:** \(R_{xy}(\tau) = R^*_{yx}(-\tau)\). Not necessarily even.

*   **Use:** Essential in communication for synchronization, channel estimation, and radar.

1.3 Convolution Theorem

• Core Principle: Convolution in the time domain is equivalent to multiplication in the frequency domain, and vice-versa.

$$x_1(t) * x_2(t) \leftrightarrow X_1(f) X_2(f)$$

$$x_1(t) x_2(t) \leftrightarrow X_1(f) * X_2(f)$$

• Application in Systems: The output \(y(t)\) of an LTI system with impulse response \(h(t)\) to input \(x(t)\) is \(y(t) = x(t) * h(t)\). In frequency domain, \(Y(f) = X(f) H(f)\), simplifying analysis.

[!TIP] Common Pitfall: Remember the duality property flips the sign in the frequency domain: \(x(t) \leftrightarrow X(f)\) implies \(X(t) \leftrightarrow x(-f)\).

2.0 AMPLITUDE MODULATION (AM) & SUPPRESSED CARRIER TECHNIQUES

2.1 Fundamentals of Modulation

• Need for Modulation:

  1. Frequency Translation: Shifts baseband signal to higher carrier frequency for efficient radiation (antenna size \(\propto \lambda\)).

  2. Multiplexing: Allows multiple signals on same channel (FDM).

  3. Noise Reduction: Certain modulation schemes (FM) offer better noise immunity.

  4. Overcomes Practical Limitations: Enables use of smaller antennas and reduces signal attenuation.

• General AM Waveform (Standard/Conventional AM):

$$s(t) = [A_c + m(t)] \cos(2\pi f_c t) = A_c[1 + \mu m_n(t)] \cos(2\pi f_c t)$$

where \(A_c\) = carrier amplitude, \(m(t)\) = message signal (assumed \( |m(t)| \leq A_m\)), \(\mu = \frac{A_m}{A_c}\) = **modulation index**, \(m_n(t) = m(t)/A_m\) = normalized message (\(|m_n(t)| \leq 1\)).

*   **Over-modulation:** \(\mu > 1\) causes envelope distortion, making envelope detection impossible.

*   **% Modulation:** \(\mu \times 100\%\).

2.2 AM Spectrum & Power Analysis

• Spectrum Derivation: Using Euler's identity:

$$s(t) = A_c \cos(2\pi f_c t) + \frac{\mu A_c}{2} [e^{j2\pi f_c t} + e^{-j2\pi f_c t}] M(f)$$

Spectrum consists of:

*   **Carrier** at \(f_c\) with amplitude \(A_c\).

*   **Upper Sideband (USB)** at \(f_c + f_m\) with amplitude \(\frac{\mu A_c}{2}|M(f)|\).

*   **Lower Sideband (LSB)** at \(f_c - f_m\) with amplitude \(\frac{\mu A_c}{2}|M(f)|\).

• Power Calculation (for single-tone modulation \(m(t)=A_m\cos(2\pi f_m t)\)):

  • Total Transmitted Power: \(P_T = P_C + P_{USB} + P_{LSB}\)

  • Carrier Power: \(P_C = \frac{A_c^2}{2R}\) (R = load resistance)

  • Sideband Power (each): \(P_{SB} = \frac{\mu^2 A_c^2}{8R} = \frac{\mu^2}{4} P_C\)

  • Total Power: \(P_T = P_C \left(1 + \frac{\mu^2}{2}\right)\)

  • Transmission Efficiency: \(\eta = \frac{P_{SB,total}}{P_T} = \frac{\mu^2}{2 + \mu^2}\)

• % Power Saving:

  • Carrier Suppressed (DSB-SC): Saves \(100 \times \frac{P_C}{P_T}\% = 100 \times \frac{1}{1+\mu^2/2}\%\)

  • One Sideband Suppressed (SSB): Saves \(100 \times \frac{P_{SB} + P_C}{P_T}\% = 100 \times \frac{1+\mu^2/4}{1+\mu^2/2}\%\)

  • Example (μ=1): Carrier save = 66.67%, One SB save = 83.33%.

[!TIP] Exam Focus: Numerical on % power saving for given μ (like μ=0.5, μ=1) is very frequent. Remember formulas: Carrier save = \(\frac{2}{2+\mu^2} \times 100\%\), One SB save = \(\frac{2+\mu^2/2}{2+\mu^2} \times 100\%\).

2.3 AM Generation Methods

• Square Law Modulator: Uses a non-linear device (diode/transistor) with a square-law transfer characteristic \(i = a_1 v + a_2 v^2\). Input \(v = A_c \cos \omega_c t + m(t)\). The \(a_2 v^2\) term generates \(A_c m(t) \cos \omega_c t\) (DSB-SC) and other terms. A bandpass filter removes DC and \(2f_c\) terms, yielding AM. Simple but inefficient.

• Balanced Modulator: Two nonlinear devices in a push-pull configuration. Carrier signals are 180° out of phase, causing carrier cancellation at output, generating pure DSB-SC without need for filtering. (Used for suppressed carrier generation).

• Envelope Detector Method: A simple diode detector (RC load) followed by a DC blocker. The diode rectifies the AM signal, and the RC filter tracks the envelope \(A_c[1+\mu m_n(t)]\). Condition for no distortion: \(RC \gg \frac{1}{\omega_c}\) (to discharge slowly) and \(RC \ll \frac{1}{\omega_m}\) (to follow message).

2.4 Demodulation of AM

• Envelope Detection: As above. Simple, cheap, used in broadcast receivers. Fails for over-modulation or low SNR.

• Synchronous (Coherent) Detection: Requires a receiver carrier \(A_c \cos(2\pi f_c t + \phi)\) phase-locked to transmitter. Multiply \(s(t)\) by this carrier and pass through LPF. Output \(\propto m(t) \cos \phi\). Needs carrier reinsertion (from a Costas loop or pilot tone). Superior performance, used in SSB/DSB-SC.

2.5 Double Sideband Suppressed Carrier (DSB-SC)

• Waveform: \(s(t) = m(t) \cos(2\pi f_c t)\). Spectrum: Two sidebands at \(f_c \pm f_m\), no carrier.

• Generation: Primarily via Balanced Modulator.

• Figure of Merit (FOM): Ratio of output SNR to input SNR for a given modulation.

  • AM (Envelope): \(FOM_{AM} = \frac{\mu^2}{2 + \mu^2}\)

  • DSB-SC (Coherent): \(FOM_{DSB-SC} = 1\)

  • Proof (DSB-SC): Input signal power \(P_s = P_m\). Input noise power \(N_i = N_0 B\). Output signal \(P_{o,s} = P_m/4\) (due to multiplication by \(\frac{1}{2}\cos\) and LPF). Output noise \(N_o = N_0 B/2\) (noise bandwidth halved). So \(FOM = \frac{P_{o,s}/N_o}{P_s/N_i} = 1\).

  • Comparison: DSB-SC has 100% higher FOM than AM for same \(P_s\) because it transmits no carrier power (which carries no information but adds to noise denominator in AM detection).

2.6 Single Sideband Suppressed Carrier (SSB-SC)

• Need: DSB-SC wastes power and bandwidth transmitting two identical sidebands. SSB transmits only one sideband, saving 50% bandwidth and 75% power (compared to AM) for single-tone.

• Generation Methods:

  1. Filter Method: Generate DSB-SC, then use a sharp bandpass filter to pass only USB or LSB. Difficult at low frequencies.

  2. Phase Shift (Hilbert Transform) Method: Uses two balanced modulators with 90° phase-shifted carriers and message signals (Hilbert transform \( \hat{m}(t) \)). Sum/difference yields SSB.

$$s_{USB}(t) = \frac{1}{2}[m(t)\cos\omega_c t - \hat{m}(t)\sin\omega_c t]$$

3.  **Phase Discrimination Method (Weaver's Method):** Modulates onto a low IF carrier, then translates to \(f_c\). Easier filtering at low IF.

• Demodulation: Requires coherent detection with a carrier reinserted at the receiver (same frequency and phase). If carrier phase error \(\phi\), output \(\propto m(t)\cos\phi + \hat{m}(t)\sin\phi\) → distortion.

2.7 Vestigial Sideband (VSB-SC)

• Need: Compromise between SSB (sharp filter needed) and DSB. Used in TV broadcast (video signal has significant low-frequency content; SSB filter would distort it).

• Generation: Transmit one full sideband + a vestige (small portion) of the other sideband. Implemented with a filter having a gradual roll-off (e.g., a trapezoidal shape in frequency domain).

• Detection: Can use envelope detection if the vestige is properly designed to cancel distortion. The receiver uses a complementary filter to restore flat overall response.

[!TIP] Exam Focus: SSB generation (Phase Shift method) and VSB application (TV) are common short notes. Remember SSB saves 50% BW and 75% power vs AM.

3.0 ANGLE MODULATION (FM & PM)

3.1 Fundamentals

• General Form: \(s(t) = A_c \cos[\theta(t)]\), where \(\theta(t) = 2\pi f_c t + \phi(t)\) is instantaneous phase.

• Frequency Modulation (FM): Instantaneous frequency \(f_i(t) = f_c + k_f m(t)\).

  • Frequency Deviation: \(\Delta f = k_f A_m\) (max shift from \(f_c\)).

  • Modulation Index: \(\beta = \frac{\Delta f}{f_m}\) (for single-tone \(m(t)=A_m\cos 2\pi f_m t\)).

  • Time-domain: \(s(t) = A_c \cos[2\pi f_c t + \beta \sin(2\pi f_m t)]\).

• Phase Modulation (PM): Instantaneous phase \(\phi(t) = k_p m(t)\).

  • Modulation Index: \(\beta = k_p A_m\).

  • Time-domain: \(s(t) = A_c \cos[2\pi f_c t + k_p m(t)]\).

  • Key Difference: In FM, \(\beta \propto 1/f_m\); in PM, \(\beta\) independent of \(f_m\).

3.2 FM Spectrum & Bandwidth

• Bessel Function Representation:

$$s(t) = A_c \sum_{n=-\infty}^{\infty} J_n(\beta) \cos[2\pi(f_c + n f_m)t]$$

Spectrum has infinite sidebands at \(f_c \pm n f_m\) with amplitudes \(A_c J_n(\beta)\).

*   **Significant Sidebands:** Those where \(|J_n(\beta)| > 0.01\) (approx). Number increases with \(\beta\).

• Carson's Rule (Practical Bandwidth):

$$\boxed{BW \approx 2(\Delta f + f_m) = 2f_m(\beta + 1)}$$

This covers ~98% of power. **Rule of thumb:** \(BW \approx 2\Delta f\) for large \(\beta\) (WBFM).

• Narrowband FM (NBFM) vs Wideband FM (WBFM):

  • NBFM: \(\beta \ll 1\) (typically < 0.5). Spectrum similar to AM (carrier + 2 sidebands). \(J_0(\beta) \approx 1\), \(J_{\pm1}(\beta) \approx \beta/2\).

  • WBFM: \(\beta \gg 1\) (typically > 1). Many sidebands, bandwidth large.

  • Proof: NBFM offers no SNR improvement over AM:

    For NBFM, \(s(t) \approx A_c \cos\omega_c t - \beta A_c m(t) \sin\omega_c t\). This is mathematically equivalent to DSB-SC with carrier. Hence, its FOM in noise is same as AM (coherent detection of DSB-SC gives FOM=1, but AM envelope gives less). Key: FM's noise advantage comes from bandwidth expansion (WBFM) and limiter action.

3.3 FM Generation Methods

• Direct Method: VCO (Voltage-Controlled Oscillator) where \(f_i \propto v_{in}(t)\). Reactance Modulator: A capacitive/inductive reactance (diode/FET) controlled by \(m(t)\) connected across tank circuit of oscillator.

• Indirect Method (Armstrong): First generate NBFM using a phase modulator (since \(\Delta \phi = \beta\) for PM). Then use frequency multipliers (non-linear circuits) to increase both \(f_c\) and \(\Delta f\) by factor \(n\), so \(\beta\) becomes \(n\beta\), achieving WBFM. Advantage: Stable carrier.

3.4 FM Demodulation Methods

• Frequency Discriminators: Convert frequency variations to amplitude variations.

  • Foster-Seeley Discriminator: Uses a tuned RF transformer (primary tuned to \(f_c\), secondaries tuned to \(f_c \pm \Delta f\)). Output across diodes is zero at \(f_c\), positive/negative for deviations. Transfer characteristic: S-shaped curve.

  • Ratio Detector: Variation of Foster-Seeley with a capacitor across load resistor. Does not require strict amplitude limiting, but limited bandwidth.

• Phase-Locked Loop (PLL) Detector: PLL in locked state acts as a perfect frequency tracker. VCO output phase \(\phi_{vco}(t)\) follows input FM phase. Differentiator (or inherent VCO control) gives output \(\propto \frac{d}{dt}\phi_{vco}(t) \propto m(t)\). Advantage: Good low-frequency response, captures signal well.

• Slope Detection: Simple FM-to-AM conversion using a tuned circuit at a sloped part of its response curve. Followed by envelope detector. Poor linearity.

3.5 Pre-emphasis & De-emphasis

• Need: In FM, noise power is uniform (white) over channel bandwidth. However, message signals (like audio) have more power at low frequencies. After demodulation, SNR is worse at high frequencies because high-frequency modulating components produce smaller frequency deviations for same amplitude (since \(\Delta f \propto m(t)\) but noise effect is constant across band).

• Solution: Artificially boost high-frequency components at transmitter (Pre-emphasis) and attenuate them at receiver (De-emphasis). The networks are complementary (RC high-pass / RC low-pass).

• SNR Improvement: For a 1st order RC network with time constant \(\tau = 1/(2\pi f_c)\) (where \(f_c\) is 3dB cutoff), the improvement at high frequencies (\(f \gg f_c\)) is:

$$\boxed{\frac{(SNR)_{out}}{(SNR)_{out,no\,emph}} \approx \left(\frac{f}{f_c}\right)^2}$$

Typical \(f_c = 2.1\) kHz (US) or 3.8 kHz (EU).

[!TIP] Exam Focus: Derivation of Carson's rule (from Bessel) and pre-emphasis SNR improvement are common. Know Foster-Seeley circuit operation and PLL as FM detector.

4.0 RECEIVER SYSTEMS

4.1 Basic Receiver Functions

• Tuning/Selectivity: Select desired signal, reject others (image, adjacent).

• Amplification: RF, IF, and Audio amplification.

• Detection/Demodulation: Extract baseband.

• Fidelity: Ability to reproduce original signal accurately (flat frequency response).

• Sensitivity: Minimum input signal power for usable output (specified SNR).

• Selectivity: Ability to separate adjacent channels.

• Automatic Volume Control (AVC/AGC): Stabilizes output amplitude for varying input signal strength.

4.2 Tuned Radio Frequency (TRF) Receiver

• Block Diagram: RF Amplifier (tuned) → Mixer (often absent) → Detector → Audio Amp. (All tuned to \(f_c\)).

• Disadvantages (Why it's obsolete):

  1. Poor Selectivity: Requires very sharp, multi-stage RF filters at high frequency → difficult/expensive.

  2. Image Frequency Problem: No frequency conversion; any signal at \(f_{image} = f_c + 2f_{IF}\) (if IF used) or just any strong signal in band passes.

  3. Instability: High-frequency amplifiers prone to oscillations.

  4. Gang Tuning: All RF stages must track accurately → mechanical complexity.

4.3 Superheterodyne Receiver (Standard AM/FM)

• Block Diagram & Function:

  1. RF Amplifier: Initial amplification, initial image rejection.

  2. Mixer: Heterodynes \(f_{RF}\) with \(f_{LO}\) to produce IF (\(f_{IF} = |f_{LO} - f_{RF}|\)). Image frequency \(f_{image} = f_{RF} + 2f_{IF}\) also produces IF.

  3. Local Oscillator (LO): Variable with tuning (usually \(f_{LO} = f_{RF} + f_{IF}\)).

  4. IF Amplifier: Heart of receiver. Fixed frequency (e.g., 455 kHz for AM, 10.7 MHz for FM). Provides most gain and primary selectivity (using fixed, high-Q filters).

  5. Detector/Discriminator: Demodulates IF signal.

  6. Audio Amplifier: Drives speaker.

  7. AGC Circuit: Derives DC from detected audio/IF to control RF/IF gain.

  8. AFC Circuit: Derives error from discriminator output to fine-tune LO frequency.

• Intermediate Frequency (IF) Choice Factors:

  • Image Rejection: Higher \(f_{IF}\) → better image rejection for same RF filter Q. But higher \(f_{IF}\) makes selectivity harder.

  • Selectivity: Lower \(f_{IF}\) → easier to achieve sharp filters (higher Q possible at lower freq).

  • Gain: Lower \(f_{IF}\) → more gain stages possible.

  • Stability: Lower \(f_{IF}\) → less prone to parasitic oscillations.

  • Standard Values: 455 kHz (AM broadcast), 10.7 MHz (FM broadcast), 38.9 MHz (TV).

• Image Frequency & Rejection Ratio (IFRR):

  • Image Frequency: \(f_{image} = f_{RF} + 2f_{IF}\) (for \(f_{LO} = f_{RF} + f_{IF}\)). It produces the same IF as the desired signal.

  • Image Frequency Rejection Ratio (IFRR): Ratio of desired signal output to image signal output (both at same IF).

$$\boxed{IFRR = \frac{V_{out,desired}}{V_{out,image}} = \sqrt{1 + Q^2 \left( \frac{2f_{IF}}{f_{RF}} \right)^2}}$$

    where \(Q\) is the Q of the RF tuned circuit. **Higher Q and higher \(f_{IF}\) improve IFRR.**

• Automatic Frequency Control (AFC): Uses a small portion of discriminator output (proportional to frequency error) to vary LO capacitance (varactor), pulling LO to correct for drift.

• Automatic Gain Control (AGC): Uses rectified detector output (DC proportional to signal strength) to control gain of RF/IF amplifiers. AGC vs AVC: AGC controls RF/IF gain automatically; AVC (Audio Volume Control) is manual user control of audio gain.

[!TIP] Exam Focus: Derivation of IFRR and factors affecting IF choice are very important. Draw and explain superhet block diagram with AFC/AGC. Know why TRF is inferior.

5.0 NOISE IN ANALOG COMMUNICATION

5.1 Noise Fundamentals

• Classification:

  • External: Atmospheric, Industrial (man-made), Cosmic.

  • Internal: Thermal (Johnson-Nyquist) noise (resistors, \(\overline{v^2} = 4kTRB\)), Shot noise (PN junctions, \(\overline{i^2} = 2qI_{DC}B\)), Flicker (1/f) noise (low freq).

• Noise Temperature (T): Equivalent temperature that would generate the same noise power in a resistor. \(N = kTB\) (Watts). Allows noise power calculation.

• Noise Figure (NF) & Noise Factor (F):

  • Noise Factor (F): Ratio of (SNR at input) / (SNR at output).

$$F = \frac{(S_i/N_i)}{(S_o/N_o)}$$

*   **Noise Figure (NF):** \(NF = 10\log_{10}(F)\) dB.

*   **Friis Formula (Cascaded Stages):** Overall noise factor dominated by first stage.

$$\boxed{F_{total} = F_1 + \frac{F_2 - 1}{G_1} + \frac{F_3 - 1}{G_1 G_2} + \cdots}$$

    where \(G_i\) = power gain of stage \(i\).

• Noise Bandwidth (\(B_N\)): Equivalent bandwidth of a filter that would pass same noise power as actual filter. For an ideal BPF of BW \(B\), \(B_N = B\). For practical filters, \(B_N > B\).

5.2 Noise Performance of Modulation Systems (Figure of Merit)

• Definition: \(FOM = \left( \frac{SNR_o}{SNR_i} \right)_{mod\, system} / \left( \frac{SNR_o}{SNR_i} \right)_{AM\, with\, coherent\, detection}\) or simply output SNR per unit bandwidth for given modulation.

  • AM (Envelope Detector): \(FOM_{AM} = \frac{\mu^2}{2 + \mu^2}\)

  • DSB-SC (Coherent): \(FOM_{DSB-SC} = 1\)

  • FM (Wideband, high \(\beta\)): \(FOM_{FM} \approx \frac{3}{2} \beta^2\)

    Derivation (simplified): For large \(\beta\), Carson BW \(B_T \approx 2\Delta f\). Output SNR \(\propto \beta^2 \times (S/N)_{i,FM}\). Since FM input SNR is \((S/N)_{i,FM} = \frac{A_c^2}{2N_0 B_T}\), after demodulation \(SNR_o \propto \beta^2 \frac{A_c^2}{2N_0}\). Hence \(FOM \propto \beta^2\).

• Comparison & Why FM is Better:

  • AM: FOM ≤ 0.5 (for μ=1). Carrier power wasted, envelope detection susceptible to noise.

  • DSB-SC: FOM = 1. Better than AM, but still linear modulation.

  • FM: FOM \(\propto \beta^2\). By increasing \(\beta\) (i.e., increasing \(\Delta f\) for same \(f_m\)), we get arbitrarily high SNR improvement at the cost of bandwidth. Also, limiter in FM receiver removes amplitude noise (AM noise) before discrimination.

[!TIP] Exam Focus: Derivation of FOM for AM and DSB-SC is mandatory. Understand why FM's FOM is much higher (\(\beta^2\) term) and the concept of threshold effect (SNR degradation when input SNR falls below a critical value due to noise causing cycle slips in discriminator).

5.3 Effect of Noise on Receivers

• AM Receiver (Envelope): Noise appears as additive disturbance on envelope. Output noise power proportional to carrier power. Threshold effect not prominent.

• FM Receiver: Noise causes two effects:

  1. AM Noise: Caused by amplitude fluctuations of IF signal. Eliminated by limiter (clips amplitude to constant).

  2. FM Noise (Angle Noise): Caused by phase/frequency fluctuations. This is the primary noise. At high input SNR, output noise is triangular spectrum. At low input SNR (threshold effect), discriminator output becomes bursty and SNR degrades rapidly. Threshold occurs when \(\Delta f / f_m \approx 1\) and input SNR is low.

6.0 SPECIAL CIRCUITS & SYSTEMS (SHORT NOTE PRIORITIES)

6.1 Phase-Locked Loop (PLL)

• Components: Phase Detector (PD) / Mixer, Low-Pass Filter (LPF), Voltage-Controlled Oscillator (VCO).

• Operation: PD output = \(K_d [\theta_{in}(t) - \theta_{vco}(t)]\). LPF removes high-freq terms. VCO output frequency \(f_{vco} = f_0 + K_v v_{ctrl}(t)\).

• States:

  • Locked: \(\theta_{in} \approx \theta_{vco}\), VCO tracks input frequency/phase. Acts as FM detector (differentiator action).

  • Hold-in Range: Max frequency offset it can track.

  • Capture Range: Range it acquires lock from free-run.

• Applications: FM demodulation, carrier recovery for DSB-SC/SSB, frequency synthesis, clock recovery.

6.2 Costas Loop

• Purpose: Demodulation of suppressed carrier signals (DSB-SC, SSB). Standard PLL fails because no carrier to lock onto.

• Modification: Two phase detectors (mixers). One with 0° LO, one with 90° LO. Their outputs (I and Q channels) are multiplied by \(\text{sgn}(I)\) and \(\text{sgn}(Q)\) (or low-pass filtered) and subtracted to generate error signal. This error drives VCO.

• Working: The loop locks to the suppressed carrier of the DSB-SC signal. The I-channel (in-phase) output after LPF is proportional to \(m(t)\). Key: The squaring operation (via multiplication) in the loop creates a strong component at \(2f_c\), allowing carrier recovery.

6.3 Balanced Modulator

• Circuit: Two identical nonlinear devices (diodes or transistors) in push-pull. Carrier applied 180° out of phase to each device. Message signal applied in parallel.

• Operation: Each device produces \(a_1(v_C \pm v_m) + a_2(v_C \pm v_m)^2\). The carrier terms in \(a_1\) cancel in differential output. The \(a_2 v_C^2\) term (carrier squared) is in-phase and cancels. The \(a_2 v_C v_m\) terms are out-of-phase and add, producing \(2a_2 v_C v_m \propto m(t)\cos\omega_c t\) (DSB-SC).

• Advantage: Good carrier suppression (>40 dB) without need for filter.

6.4 Frequency Discriminators

• Foster-Seeley Discriminator:

  • Circuit: Primary coil tuned to \(f_c\), two secondary coils (tuned to \(f_c \pm \Delta f\)) connected to diodes in series-opposing. Load resistor across diodes.

  • Operation: At \(f_c\), voltages across secondaries are equal and opposite → zero output. For \(f > f_c\), upper secondary voltage > lower → positive output. For \(f < f_c\), negative output. Transfer characteristic is linear over small \(\Delta f\).

• Ratio Detector: Similar but with a capacitor across load resistor. Does not require limiting before it. Output is always positive (rectified). Limited bandwidth.

6.5 Key System Parameters

• Image Signal Rejection: Critical in superhet receivers. If poor, strong off-channel signal at \(f_{image}\) will down-convert to same IF and interfere. Improved by: higher RF Q, higher \(f_{IF}\), better RF filtering.

• Selectivity vs. Fidelity: Trade-off. High selectivity (sharp filters) removes adjacent channel interference but can distort message (especially if message has high-frequency components), reducing fidelity. Broadcast receivers balance this.

• AGC (Automatic Gain Control): Maintains constant output level despite input signal variations. Uses rectified detector output to control gain of RF/IF amplifiers via reverse-biased diodes or transistor bias. Prevents overloading and fading.

• AFC (Automatic Frequency Control): Corrects LO frequency drift. Uses DC from discriminator (proportional to frequency error) to adjust LO varactor. Keeps receiver tuned to signal.

• IF Frequency: 455 kHz (AM): Compromise between image rejection (need high) and selectivity/gain (need low). 10.7 MHz (FM): High enough for good image rejection (since FM broadcast band is 88-108 MHz), and allows use of ceramic filters with good selectivity.

6.6 Modulation Scheme Applications

[!TIP] Exam Focus: Be ready to draw Costas loop, balanced modulator, Foster-Seeley discriminator. Know applications table. Understand the fundamental trade-offs (selectivity/fidelity, image rejection/IF choice).