Analog Communication (EC-403) - Unit 2 Short Notes

UNIT 2: ANALOG COMMUNICATION - EXAM-FOCUSED SHORT NOTES

I. SIGNAL ANALYSIS & FOURIER TRANSFORM (FOUNDATION)

Fourier Transform (FT) Properties & Theorems

• Definition: The Fourier Transform $X(f)$ of a continuous-time signal $x(t)$ is:

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

The inverse transform is:

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

[!TIP] Exam Focus: Derive time/frequency shifting and convolution theorem. Duality is often asked for standard signals.

Fourier Transform of Standard Signals

[!TIP] Derivation Must: FT of Gate function $\text{rect}(t/\tau)$ is a classic exam question. Practice the integration.

Correlation Functions

• Auto-correlation $$\displaystyle R_{xx}(\tau) $$: Measure of similarity between a signal and its time-shifted version.

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

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

*   **Properties:** Even function ($$\displaystyle R_{xx}(-\tau)=R_{xx}(\tau) $$), max at $$\displaystyle \tau=0 $$ ($$\displaystyle R_{xx}(0)=E $$ or $P$).

*   **Relation to ESD/PSD:** **Wiener-Khinchin Theorem:** The Fourier Transform of $$\displaystyle R_{xx}(\tau) $$ is the **Energy Spectral Density (ESD)** $\Psi(f)$ for energy signals, or **Power Spectral Density (PSD)** $$\displaystyle S_{xx}(f) $$ for power signals.

$$\Psi(f) = |X(f)|^2, \quad S_{xx}(f) = \lim_{T\to\infty} \frac{|X_T(f)|^2}{T}$$

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

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

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

*   **Relation to CSD:** Fourier Transform of $$\displaystyle R_{xy}(\tau) $$ is **Cross Spectral Density (CSD)** $$\displaystyle S_{xy}(f) $$.

[!TIP] Key Distinction: ESD $$\displaystyle |X(f)|^2 $$ for finite energy signals. PSD $S(f)$ for finite power signals (limit process). Auto-correlation and PSD are Fourier pairs.

II. AMPLITUDE MODULATION (AM) & ITS VARIANTS

Need & Principle of Modulation

• Basic Block Diagram: Message Source → Transmitter (Modulator) → Channel → Receiver (Demodulator) → Message Destination

• Main Advantages:

  1. Frequency Translation: Shifts baseband spectrum to higher carrier frequency → practical antenna size ($l \propto \lambda$).

  2. Multiplexing: Different signals on different carriers → efficient channel use.

  3. Noise Reduction: Certain modulation types (FM) offer improved SNR.

  4. Overcoming Hardware Limitations: Allows design of optimized circuits at carrier frequency.

Conventional AM (DSB-FC)

• Time-domain Equation:

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

where $$\displaystyle A_c $$ = carrier amplitude, $m(t)$ = message (normalized $|m(t)| \leq 1$), $$\displaystyle k_a $$ = amplitude sensitivity.

**Modulation Index:** $$\displaystyle m = k_a \max|m(t)| $$, $0 \leq m \leq 1$ for no overmodulation.

**% Modulation** $$\displaystyle = m \times 100\% $$.

• Spectrum & Bandwidth:

$$S(f) = \frac{A_c}{2}[\delta(f-f_c) + \delta(f+f_c)] + \frac{A_c k_a}{2}[M(f-f_c) + M(f+f_c)]$$

**Bandwidth** $$\displaystyle B_T = 2f_m $$ (where $$\displaystyle f_m $$ = max modulating frequency).

• Power Distribution:

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

  • Total Power: $$\displaystyle P_T = P_c (1 + \frac{m^2}{2}) $$

  • Sideband Power: $$\displaystyle P_{SB} = P_T - P_c = P_c \frac{m^2}{2} $$

  • Power Efficiency $$\displaystyle \eta = \frac{P_{SB}}{P_T} = \frac{m^2}{2+m^2} $$. Max $$\displaystyle \eta = 33.33\% $$ at $$\displaystyle m=1 $$.

• Generation:

  • Square Law Modulator: Non-linear device (diode) with $$\displaystyle i = a_1 v + a_2 v^2 $$. Input $$\displaystyle v = A_c \cos \omega_c t + m(t) $$. Squaring term generates $$\displaystyle 2A_c m(t) \cos \omega_c t $$ (DSB-SC) and $$\displaystyle m^2(t) $$ term. Filter out $$\displaystyle m^2(t) $$ & carrier to get AM? Actually, square law alone gives AM with carrier if bias is set correctly.

  • Envelope Detector (Demodulation): Simple diode + RC filter. Works if $$\displaystyle m < 1 $$ and $RC$ time constant satisfies $$\displaystyle \frac{1}{\omega_c} \ll RC \ll \frac{1}{\omega_m} $$.

• Demodulation Techniques:

  • Envelope Detection: As above. Simple, cheap.

  • Synchronous/Coherent Detection: Multiply $s(t)$ by locally generated carrier $$\displaystyle \cos(2\pi f_c t + \phi) $$ (phase-locked), then LPF. Requires carrier phase sync. Better performance in noise.

Double Sideband Suppressed Carrier (DSB-SC)

• Equation: $$\displaystyle s(t) = A_c m(t) \cos(2\pi f_c t) $$

• Spectrum: $$\displaystyle S(f) = \frac{A_c}{2}[M(f-f_c) + M(f+f_c)] $$. No carrier impulse.

• Generation: Balanced Modulator (e.g., diode ring). Two carriers 180° out of phase → carrier cancels at output → only DSB-SC.

• Demodulation: Coherent Detection only. Multiply by synchronized carrier, LPF → $$\displaystyle \frac{A_c}{2} m(t) $$.

• Figure of Merit (FOM) for DSB-SC:

  FOM $$\displaystyle = \frac{(SNR)_{out}}{(SNR)_{c,in}} $$ for coherent detection.

  For DSB-SC with coherent detection: $$\displaystyle (SNR)_{out} = \frac{A_c^2 P_m}{4 N_0 B} $$ (for AWGN). Carrier power $$\displaystyle P_c = \frac{A_c^2}{2R} $$.

$$\boxed{\text{FOM}_{\text{DSB-SC}} = \frac{P_m}{2 N_0 B}}$$

where $$\displaystyle P_m $$ = message power, $$\displaystyle N_0/2 $$ = noise PSD, $B$ = bandwidth ($$\displaystyle 2f_m $$).

Single Sideband Suppressed Carrier (SSB-SC)

• Principle: Transmit only one sideband (USB or LSB) + suppressed carrier. Bandwidth $$\displaystyle = f_m $$ (half of DSB/AM).

• Generation Methods (Exam Focus):

  1. Phase Discrimination / Filter Method:

     • Generate DSB-SC using balanced modulator.

     • Use a very sharp sideband filter to pass only USB or LSB.

     • Block Diagram: Message → Hilbert Transform → 90° Phase Shift → Balanced Modulator (with carrier) → Sideband Filter → SSB Output.

  2. Hilbert Transform Method (Conceptual): Use $m(t)$ and its Hilbert transform $\hat{m}(t)$ (90° phase-shifted all frequencies) to generate SSB directly: $$\displaystyle s(t) = A_c [m(t) \cos \omega_c t \mp \hat{m}(t) \sin \omega_c t] $$.

• Demodulation: Coherent detection with locally generated carrier (needs phase sync).

• Power Saving Calculation:

  • In AM (DSB-FC): $$\displaystyle P_{total} = P_c (1 + \frac{m^2}{2}) $$

  • In SSB-SC: $$\displaystyle P_{total} = \frac{A_c^2}{4R} P_m $$ (carrier suppressed, one sideband suppressed).

  • % Power Saving compared to AM:

$$\boxed{\text{% Saving} = \left(1 - \frac{P_{SSB}}{P_{AM}}\right) \times 100\%}$$

For $$\displaystyle m=1 $$ (100% modulation AM): $$\displaystyle P_{AM} = 1.5 P_c $$, $$\displaystyle P_{SSB} = \frac{P_c}{2} P_m $$ (if $$\displaystyle P_m=1 $$ for normalized $m(t)$). Then % Saving $$\displaystyle = \left(1 - \frac{0.5 P_c}{1.5 P_c}\right) \times 100\% = 66.67\% $$.

**General Formula for given $m$:**

$$\boxed{\text{% Saving} = \frac{2 + m^2}{2 + 2m^2} \times 100\%}$$

Vestigial Sideband (VSB-SC)

• Need: SSB requires very sharp filters (impractical for video signals with DC). VSB is a compromise: one full sideband + a vestige (small part) of the other sideband.

• Bandwidth: $$\displaystyle B_{VSB} = f_m + f_v $$ ($$\displaystyle f_v $$ = vestige width), between DSB ($$\displaystyle 2f_m $$) and SSB ($$\displaystyle f_m $$).

• Generation & Detection: Use a vestigial filter at transmitter (non-symmetrical about $$\displaystyle f_c $$) and a compensating filter at receiver (symmetrical) to restore original DSB shape. Carrier is transmitted (unlike SSB). Used in TV broadcasting (NTSC, PAL).

Comparison of AM, DSB-SC, SSB-SC

III. ANGLE MODULATION (FM & PM)

Basic Concepts

• Definition: Information is conveyed by varying the instantaneous angle $\theta(t)$ of the carrier.

$$s(t) = A_c \cos[\theta(t)] = A_c \cos(\omega_c t + \phi(t))$$

where $\phi(t)$ = instantaneous phase deviation.

• Frequency Modulation (FM): $$\displaystyle \phi(t) = k_f \int_{-\infty}^{t} m(\tau) d\tau $$

  Instantaneous Frequency: $$\displaystyle f_i(t) = f_c + \Delta f(t) $$, where $$\displaystyle \Delta f(t) = k_f m(t) $$.

  Frequency Deviation: $$\displaystyle \Delta f = k_f \max|m(t)| $$.

  Modulation Index: $$\displaystyle \beta = \frac{\Delta f}{f_m} = \frac{k_f A_m}{f_m} $$ (for sinusoidal $$\displaystyle m(t)=A_m \sin 2\pi f_m t $$).

• Phase Modulation (PM): $$\displaystyle \phi(t) = k_p m(t) $$

  Instantaneous Frequency: $$\displaystyle f_i(t) = f_c + \frac{1}{2\pi} \frac{d\phi(t)}{dt} = f_c + \frac{k_p}{2\pi} \frac{dm(t)}{dt} $$.

  Phase Deviation: $$\displaystyle \Delta \phi = k_p A_m $$.

  Modulation Index: $$\displaystyle \beta_p = k_p A_m $$ (for sinusoidal $m(t)$).

  Key Difference: In FM, $$\displaystyle \beta \propto \frac{A_m}{f_m} $$; in PM, $$\displaystyle \beta_p \propto A_m $$ only.

Narrowband FM (NBFM) vs. Wideband FM

• NBFM Approximation ($\beta \ll 1$):

  Using $$\displaystyle \cos(\omega_c t + \beta \sin \omega_m t) \approx \cos \omega_c t - \beta \sin \omega_m t \sin \omega_c t $$

$$s_{NBFM}(t) \approx A_c \cos \omega_c t - \frac{A_c \beta}{2} [\cos(\omega_c-\omega_m)t - \cos(\omega_c+\omega_m)t]$$

*   **Similar to DSB-SC** with carrier and two sidebands. Bandwidth $$\displaystyle = 2f_m $$.

*   **SNR Performance:** NBFM offers **no SNR improvement** over DSB-SC (coherent) or AM (coherent). Proof: Both have output SNR $$\displaystyle \propto \frac{A_c^2 P_m}{N_0 B} $$.

• Wideband FM ($$\displaystyle \beta > 1 $$):

  • Carson's Rule (Bandwidth):

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

*   **Noise Immunity:** FM's **threshold effect** and **capture effect** provide superior noise performance over AM for $$\displaystyle \beta > 1 $$.

Generation of FM

• Direct Method (Varactor Diode Modulator):

  • Principle: Varying capacitance of a varactor diode (reverse-biased) with $m(t)$ changes the resonant frequency of an LC oscillator → directly generates FM.

  • Circuit: Message Amp → Varactor Diode in parallel with LC Tank of Oscillator.

• Indirect Method (Armstrong Method):

  • Principle: First generate NBFM using a phase modulator (balanced modulator with 90° phase-shifted carrier). Then frequency multiply (using non-linear device) to increase $\beta$ and $$\displaystyle f_c $$ to desired values.

  • Block: Message → Integrator → Phase Modulator (with 90° shifted carrier) → Frequency Multiplier → FM Output.

Demodulation of FM (Very High Frequency)

• Frequency Discriminators: Convert frequency variations to amplitude variations.

  1. Foster-Seeley Discriminator:

     • Circuit: Tuned transformer (primary: RF input; secondary: two tuned circuits at $$\displaystyle f_c $$, one slightly above, one slightly below) followed by diode envelope detectors.

     • Operation: At $$\displaystyle f_c $$, secondary voltages are 90° out of phase → sum zero. Deviation $\pm \Delta f$ → one diode output > other → positive/negative DC output.

  2. Ratio Detector: Variation of Foster-Seeley with a capacitor across the load resistor → eliminates need for strict amplitude limiting.

• Phase-Locked Loop (PLL) Detector:

  • Components: Phase Detector (PD), Loop Filter (LF), Voltage-Controlled Oscillator (VCO).

  • Operation: FM signal → PD (with VCO output) → error voltage proportional to frequency deviation → LF smoothes → controls VCO frequency. VCO control voltage is the demodulated output. Excellent tracking, low noise.

• Slope Detector: Simple tuned circuit at slope of its response curve → FM to AM → envelope detector. Poor linearity.

Carson's Rule

• Statement: For a wideband FM signal with modulation index $\beta$ and max modulating frequency $$\displaystyle f_m $$, the required transmission bandwidth is approximately:

$$\boxed{B_T \approx 2(\Delta f + f_m)}$$

where $$\displaystyle \Delta f = \beta f_m $$ is peak frequency deviation.

• Significance: Contains ~98% of total signal power. Used for channel allocation.

• Numerical Problem: Given $$\displaystyle s(t) = A_c \cos(2\pi f_c t + \beta \sin 2\pi f_m t) $$, identify $\beta$, $$\displaystyle f_m $$, $$\displaystyle \Delta f = \beta f_m $$, then $$\displaystyle B_T = 2(\Delta f + f_m) $$.

Pre-emphasis & De-emphasis (Very High Frequency)

• Significance: Noise in FM receivers has a high-frequency bias (noise PSD often $$\displaystyle \propto f^2 $$). To improve output SNR for high-frequency modulating signals, boost high frequencies at transmitter (pre-emphasis) and attenuate them equally at receiver (de-emphasis). Net effect: flat signal spectrum, reduced noise in high-frequency band.

• Circuit Implementation: Simple RC networks.

  • Pre-emphasis Filter (Transmitter): High-pass filter. Transfer function $$\displaystyle H_{pre}(f) = 1 + j\frac{f}{f_x} $$ ($$\displaystyle f_x $$ = break frequency, e.g., 2122 Hz for FCC).

  • De-emphasis Filter (Receiver): Low-pass filter (complementary). $$\displaystyle H_{de}(f) = \frac{1}{1 + j\frac{f}{f_x}} $$.

  • Net: $$\displaystyle H_{pre}(f) H_{de}(f) = 1 $$ for signal. Noise at high $f$ is attenuated by $$\displaystyle |H_{de}(f)|^2 $$.

• Neat Sketch: Show message $m(t)$ → Pre-emphasis (HPF) → FM Modulator → Channel → FM Demodulator → De-emphasis (LPF) → Output $m(t)$.

[!TIP] Exam Focus: Derive how pre-emphasis/de-emphasis improves SNR. Know the RC circuit values and break frequency.

IV. NOISE IN ANALOG COMMUNICATION SYSTEMS

Noise Fundamentals

• Definition: Unwanted random electrical signals that interfere with the desired signal.

• Classification:

  • External Noise: Atmospheric, Industrial, Galactic (extraterrestrial).

  • Internal Noise: Generated within the system/receiver. Most critical: Thermal (Johnson-Nyquist), Shot, Flicker ($1/f$), Transit-time.

• Key Parameters:

  • Noise Temperature $$\displaystyle T_n $$: Temperature (in Kelvin) that would produce the same available noise power as the actual noise source. $$\displaystyle P_n = k T_n B $$ ($k$ = Boltzmann's constant).

  • Noise Figure $F$: Ratio of output SNR to input SNR (for a given bandwidth). $$\displaystyle F = \frac{(SNR)_{in}}{(SNR)_{out}} $$. In dB: $$\displaystyle NF = 10 \log_{10} F $$. $F \geq 1$ (0 dB).

  • Noise Bandwidth $$\displaystyle B_n $$: Bandwidth of an ideal brick-wall filter that passes same noise power as the actual filter with transfer function $$\displaystyle |H(f)|^2 $$. $$\displaystyle B_n = \frac{\int_0^\infty |H(f)|^2 df}{|H(f_c)|^2} $$.

Noise in AM Systems

• Effect: Noise appears at demodulator output directly (envelope detector) or after multiplication (coherent detector).

• Figure of Merit (FOM) for Coherent AM Detection:

  For AM with modulation index $m$, carrier power $$\displaystyle P_c $$, message power $$\displaystyle P_m $$.

  Output SNR (coherent) $$\displaystyle \propto \frac{A_c^2 P_m m^2}{4 N_0 B} $$.

  Input SNR (to modulator) $$\displaystyle \propto \frac{A_c^2}{2 N_0 B} $$ (carrier power in noise bandwidth).

$$\boxed{\text{FOM}_{\text{AM-coherent}} = \frac{m^2}{2}}$$

*   **Comparison:** For $$\displaystyle m=1 $$, FOM = 0.5. **DSB-SC FOM = 1** (from Unit 2, Sec II). **SSB-SC FOM = 1** (same as DSB-SC but with half bandwidth).

*   **Conclusion:** **SSB/DSB-SC (coherent) > AM (coherent) > AM (envelope)** in noise performance for same transmitted power.

Noise in FM Systems

• Threshold Effect: For low SNR at FM receiver input, discriminator output SNR degrades rapidly (non-linear region). Below a threshold (typically 10-15 dB), performance collapses.

• Capture Effect: In FM, when multiple signals arrive, the strongest one captures the receiver, suppressing weaker ones (unlike AM where they add).

• Why FM is More Immune?

  1. Constant Amplitude: Limiter in FM receiver removes amplitude noise → only frequency deviations matter.

  2. Wideband: Carson's rule bandwidth $$\displaystyle >> $$ message bandwidth → noise power spread over larger BW → less noise in message band.

  3. SNR Improvement: For high $\beta$ (wideband FM), output SNR $$\displaystyle \propto \beta^2 $$ (Carson's rule: $$\displaystyle B_T \propto \beta $$, noise power $$\displaystyle \propto B_T $$, signal power $$\displaystyle \propto \beta^2 $$). So $$\displaystyle (SNR)_{out} \propto \beta^2 / \beta = \beta $$.

  FOM for FM (high $\beta$): $$\displaystyle \text{FOM}_{FM} \approx \frac{3}{2} \beta^2 $$ (conceptual, compared to AM's $$\displaystyle m^2/2 $$). For $$\displaystyle \beta=5 $$, FOM ~37.5 vs AM's 0.5 → huge improvement.

V. RECEIVERS & ASSOCIATED CIRCUITS

TRF (Tuned Radio Frequency) Receiver

• Block Diagram: RF Amplifier (tuned) → Mixer (no, actually TRF has multiple RF amps & detectors) → Detector → AF Amplifier. All stages tuned to carrier frequency.

• Operation: Each stage (RF amps, detector) is tuned to the desired station's RF. Direct conversion.

• Disadvantages/Limitations:

  1. Selectivity: Hard to achieve high selectivity at high RF (requires high-Q tuned circuits, unstable).

  2. Image Frequency: No image rejection (no frequency conversion). Any signal at $$\displaystyle f_{image} = f_c + 2f_{IF} $$ (if IF existed) or just any strong signal at nearby frequency causes interference.

  3. Gain & Stability: Gain must be high at RF → prone to oscillations, instability.

  4. Bandwidth: Bandwidth of tuned circuits must be wide enough for entire AM band (550-1600 kHz) → poor selectivity.

  Why Superhet replaced TRF: Superhet converts to fixed, lower IF → high, stable selectivity & gain at IF.

Superheterodyne Receiver (Very High Frequency)

• Detailed Block Diagram & Function:

  1. RF Amplifier: Tuned to $$\displaystyle f_c $$, amplifies signal, provides initial image rejection.

  2. Mixer: Multiplies RF input with Local Oscillator (LO) signal. Produces sum & difference frequencies.

  3. Local Oscillator (LO): Generates $$\displaystyle f_{LO} = f_c + f_{IF} $$ (high-side injection) or $$\displaystyle f_{LO} = f_c - f_{IF} $$ (low-side). Must be stable & tunable.

  4. IF Amplifier: Core of receiver. Tuned to fixed $$\displaystyle f_{IF} $$ (e.g., 455 kHz for AM, 10.7 MHz for FM). Provides most gain and high selectivity (using ceramic/SAW filters).

  5. Detector/Demodulator: AM: envelope detector or product detector. FM: discriminator or PLL.

  6. AF Amplifier: Amplifies audio to drive speaker.

  7. Automatic Gain Control (AGC): Derived from detector output → controls gain of RF/IF amps to maintain constant output level.

• Image Frequency:

  • Definition: An undesired input frequency $$\displaystyle f_{img} $$ that, after mixing with LO, produces the same IF as the desired signal.

  • How it Arises: For desired $$\displaystyle f_c $$, $$\displaystyle f_{IF} = |f_{LO} - f_c| $$. If another signal $$\displaystyle f_{img} $$ exists such that $$\displaystyle |f_{LO} - f_{img}| = f_{IF} $$, it also gets converted to IF.

  • Calculation: For high-side injection ($$\displaystyle f_{LO} = f_c + f_{IF} $$):

$$f_{img} = f_{LO} + f_{IF} = (f_c + f_{IF}) + f_{IF} = f_c + 2f_{IF}$$

*   **Image Frequency Rejection Ratio (IFRR):** Ratio of desired signal gain at $$\displaystyle f_c $$ to undesired signal gain at $$\displaystyle f_{img} $$.

$$\boxed{\text{IFRR} = \frac{G(f_c)}{G(f_{img})} \approx Q_{RF} \cdot \frac{f_c}{f_{IF}}}$$

    where $$\displaystyle Q_{RF} $$ = Q-factor of RF amplifier tuned circuit. **Higher RF Q → better image rejection.**

• Choice of Intermediate Frequency (IF):

  • Low IF (e.g., 455 kHz): Easier to achieve high selectivity (higher Q possible), lower cost. But image frequency problem is worse ($$\displaystyle f_{img} = f_c + 910 $$ kHz, very close).

  • High IF (e.g., 10.7 MHz for FM): Easier image rejection ($$\displaystyle f_{img} $$ far away), better adjacent channel rejection. But selectivity harder at high frequency (lower Q), more expensive.

  • Compromise: For AM, 455 kHz is standard. For FM, 10.7 MHz is standard.

Automatic Gain Control (AGC) / Automatic Volume Control (AVC)

• Need: Maintain constant audio output level despite variations in received signal strength (fading, distance).

• Principle: Detect average signal level (from detector output) → generate DC control voltage → apply to gain-controlled stages (usually IF amps, sometimes RF).

• Block Diagram: Detector Output → AGC Filter (LPF) → AGC Amplifier → Gain-Controlled IF Amplifier.

• Difference AGC vs AFC:

  • AGC: Controls amplitude (gain) to stabilize output level.

  • AFC (Automatic Frequency Control): Controls frequency of LO to correct for drift. Uses discriminator output (which is proportional to frequency error) → filtered → applied to LO varactor.

Selectivity & Fidelity

• Selectivity: Ability of receiver to separate desired signal from adjacent channel signals. Determined by IF filter bandwidth. High selectivity → narrow IF BW.

• Fidelity: Ability to accurately reproduce the original message signal (audio). Requires sufficient bandwidth to pass all modulating frequencies. High fidelity → wide IF BW.

• Trade-off: Increasing selectivity (narrow BW) reduces fidelity (distorts high audio freqs). AM receivers compromise (BW ~5-10 kHz).

VI. SPECIAL CIRCUITS & CONCEPTS (FREQUENT SHORT NOTES)

Phase-Locked Loop (PLL)

• Basic Components:

  1. Phase Detector (PD): Output voltage $$\displaystyle v_d(t) \propto \sin[\phi_{in}(t) - \phi_{VCO}(t)] $$.

  2. Loop Filter (LF): Low-pass filter. Removes high-freq PD output & controls loop dynamics.

  3. Voltage-Controlled Oscillator (VCO): Output frequency $$\displaystyle f_{VCO} = f_0 + K_v v_f(t) $$, where $$\displaystyle v_f(t) $$ is filtered PD output.

• Operation as FM Detector:

  • FM signal (varying $$\displaystyle f_i $$) → PD (with VCO output).

  • If VCO tracks input, $$\displaystyle v_f(t) $$ becomes a voltage proportional to frequency deviation → demodulated output.

  • Advantages: High gain, good linearity, inherent noise filtering.

Costas Loop

• Specific PLL configuration for demodulating DSB-SC/SSB (suppressed carrier signals).

• Diagram: Input (DSB-SC) → Split → Two PDs:

  • PD1: Input × VCO output (0°)

  • PD2: Input × VCO output 90° phase-shifted

  • Both PD outputs → LF → controls VCO.

  • Output: The 0° PD output (after LPF) is the demodulated baseband $m(t)$. The 90° PD output provides error signal for carrier recovery (phase lock).

• Why needed for DSB-SC? No carrier to synchronize to. Costas loop extracts and locks to the suppressed carrier phase.

Balanced Modulator

• Circuit: Typically a diode ring or transistor differential pair.

• Operation: Two carriers 180° out of phase applied to two arms. Desired: carrier cancellation → output contains only $$\displaystyle m(t) \cos \omega_c t $$ (DSB-SC). Used for generating DSB-SC or AM with suppressed carrier.

FM Transmitter Block Diagram

Message Signal → Pre-emphasis → Direct/Indirect FM Modulator → Power Amplifier → Antenna

• Direct FM: Pre-emphasis → Varactor Modulator (in VCO) → Power Amp.

• Indirect FM: Pre-emphasis → Integrator → Phase Modulator (with 90° shifted carrier) → Frequency Multiplier → Power Amp.

Convolution Theorem

• Statement: The Fourier Transform of the convolution of two time-domain signals is the product of their individual Fourier Transforms.

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

• Significance: Fundamental property of LTI systems. If $x(t)$ is input, $h(t)$ is impulse response, then output $$\displaystyle y(t) = x(t)*h(t) \leftrightarrow Y(f) = X(f)H(f) $$. Simplifies analysis: multiplication in freq domain ↔ convolution in time domain.

[!TIP] Final Exam Strategy: For short notes, draw diagrams where possible (PLL, Costas Loop, Superhet, Discriminators). Box key formulas (Carson's Rule, FOM, IFRR, Power Saving). Compare AM schemes in a table. Derive FT of gate function and FOM for DSB-SC step-by-step.