Analog Communication (EC-403) - Unit 1 Short Notes

1. Fourier Transform & Signal Analysis

Properties of Fourier Transform

• Time shifting: $$\displaystyle x(t - t_0) \leftrightarrow X(f) e^{-j2\pi f t_0} $$

• Frequency shifting: $$\displaystyle x(t) e^{j2\pi f_0 t} \leftrightarrow X(f - f_0) $$

• Time scaling: $$\displaystyle x(at) \leftrightarrow \frac{1}{|a|} X\left(\frac{f}{a}\right) $$

• Frequency scaling: $$\displaystyle X(af) \leftrightarrow \frac{1}{|a|} x\left(\frac{t}{a}\right) $$

• Duality: If $$\displaystyle x(t) \leftrightarrow X(f) $$, then $$\displaystyle X(t) \leftrightarrow x(-f) $$

• Differentiation: $$\displaystyle \frac{d^n x(t)}{dt^n} \leftrightarrow (j2\pi f)^n X(f) $$

• Integration: $$\displaystyle \int_{-\infty}^{t} x(\tau) d\tau \leftrightarrow \frac{X(f)}{j2\pi f} + \frac{1}{2} X(0) \delta(f) $$

• Parseval's theorem: $$\displaystyle \int_{-\infty}^{\infty} |x(t)|^2 dt = \int_{-\infty}^{\infty} |X(f)|^2 df $$ (energy signals)

Fourier Transform of Standard Signals

Convolution Theorem

• Time convolution ↔ Frequency multiplication:

  $$\displaystyle x(t) * h(t) \leftrightarrow X(f) \cdot H(f) $$

• Frequency convolution ↔ Time multiplication:

  $$\displaystyle x(t) \cdot h(t) \leftrightarrow \frac{1}{2\pi} X(f) * H(f) $$

Correlation Functions

• Auto-correlation (energy signal):

  $$\displaystyle R_{xx}(\tau) = \int_{-\infty}^{\infty} x(t) x^*(t+\tau) dt $$

  Properties: Even function, maximum at $$\displaystyle \tau=0 $$, $$\displaystyle R_{xx}(0) = $$ total energy.

• Cross-correlation:

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

• Energy Spectral Density (ESD):

  $$\displaystyle E_x(f) = |X(f)|^2 $$. Properties: Non-negative, even, $$\displaystyle \int_{-\infty}^{\infty} E_x(f) df = $$ total energy.

• Power Spectral Density (PSD):

  $$\displaystyle S_x(f) = \lim_{T\to\infty} \frac{1}{T} |X_T(f)|^2 $$, where $$\displaystyle X_T(f) $$ is FT over $[-T/2, T/2]$.

  Properties: Non-negative, even, $$\displaystyle \int_{-\infty}^{\infty} S_x(f) df = $$ total power.

• Wiener-Khinchin theorem: For power signals, $$\displaystyle S_x(f) = \mathcal{F}\{R_{xx}(\tau)\} $$.

[!TIP]

Exam Focus: Fourier transforms of gate, impulse, step, and signum are frequently asked. Practice piecewise signal transforms (e.g., Jun 2024). Remember duality: $$\displaystyle \mathcal{F}\{X(t)\} = x(-f) $$.

2. Amplitude Modulation (AM) Techniques

Need & Principles of Modulation

• Basic communication system:

  DiagramCANVAS: Source → Transmitter ( modulator + carrier ) → Channel → Receiver (demodulator) → Destination

• Advantages:

  • Frequency translation for efficient antenna radiation ($\lambda/4$).

  • Multiplexing: multiple signals on different carriers.

  • Noise immunity (some schemes).

  • Overcomes limitations of low-frequency transmission.

Conventional AM (DSB-FC)

• Time-domain equation:

  $$\displaystyle s(t) = A_c [1 + m_a \cos(2\pi f_m t)] \cos(2\pi f_c t) $$

  where $$\displaystyle m_a = \frac{V_m}{A_c} $$ is modulation index ($$\displaystyle 0 \leq m_a \leq 1 $$).

• Spectrum:

  Carrier at $$\displaystyle f_c $$, Upper Sideband (USB) at $$\displaystyle f_c+f_m $$, Lower Sideband (LSB) at $$\displaystyle f_c-f_m $$.

  Amplitudes: Carrier $$\displaystyle A_c $$, each sideband $$\displaystyle \frac{m_a A_c}{2} $$.

• Over-modulation: $$\displaystyle m_a > 1 $$ causes envelope distortion.

• Power calculations (load resistance $R$):

  • Carrier power: $$\displaystyle P_c = \frac{A_c^2}{2R} $$

  • Each sideband power: $$\displaystyle P_{USB} = P_{LSB} = \frac{m_a^2 A_c^2}{4R} = \frac{m_a^2}{2} P_c $$

  • Total power: $$\displaystyle P_t = P_c \left(1 + \frac{m_a^2}{2}\right) $$

Suppressed Carrier Variants

• DSB-SC:

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

  • Spectrum: Only USB and LSB (no carrier).

  • Generation: Balanced modulator (product detector) or square-law modulator with filtering.

  • Demodulation: Synchronous detector (coherent detection).

• SSB-SC:

  • Generation:

    • Filter method: Generate DSB-SC, then filter out one sideband.

    • Phase discrimination (Hilbert transform):

      $$\displaystyle s(t) = m(t) \cos(2\pi f_c t) \mp m_h(t) \sin(2\pi f_c t) $$,

      where $$\displaystyle m_h(t) $$ is Hilbert transform of $m(t)$.

  • Demodulation: Synchronous detection with carrier reinsertion.

  • Bandwidth: $$\displaystyle f_m $$ (half of DSB), Power efficiency: Higher than DSB-SC (only one sideband transmitted).

• VSB-SC:

  • Need: Practical filtering when message has significant low-frequency content (e.g., TV video). Full SSB filtering near carrier is difficult.

  • Generation: Filter DSB-SC with a vestigial filter that passes one sideband fully and a vestige of the other.

  • Detection: Coherent detection (carrier suppressed). If carrier is partially transmitted (as in TV), envelope detection possible.

Modulation/Demodulation Circuits

• Square-law modulator: Uses nonlinear device (diode). Output: $$\displaystyle i = a_0 + a_1 v + a_2 v^2 $$. With carrier $$\displaystyle v_c $$ and message $$\displaystyle v_m $$, product term generates DSB-FC.

• Balanced modulator: Two nonlinear devices with carrier $$\displaystyle 180^\circ $$ out of phase; subtract outputs to cancel carrier → DSB-SC.

• Envelope detector: Diode + RC filter. Conditions for distortionless detection:

  • $$\displaystyle m_a \leq 1 $$

  • $$\displaystyle \frac{1}{\omega_c} \ll RC \ll \frac{1}{\omega_m} $$, where $$\displaystyle \omega_m $$ is max modulating frequency.

• Synchronous detector: Multiply by coherent carrier, then LPF. Requires carrier phase synchronization.

Performance Comparison

Power Saving in AM Systems

For AM with modulation index $$\displaystyle m_a $$:

• Total power: $$\displaystyle P_t = P_c \left(1 + \frac{m_a^2}{2}\right) $$

• Carrier suppression saving:

  $$\displaystyle \% = \frac{P_c}{P_t} \times 100 = \boxed{\frac{1}{1 + \frac{m_a^2}{2}} \times 100} $$

• One sideband suppression saving:

  $$\displaystyle \% = \frac{P_c + P_{LSB}}{P_t} \times 100 = \boxed{\frac{1 + \frac{m_a^2}{4}}{1 + \frac{m_a^2}{2}} \times 100} $$

[!TIP]

Common Pitfall: Modulation index from antenna currents: $$\displaystyle I_t = I_c \sqrt{1 + \frac{m_a^2}{2}} $$. Solve for $$\displaystyle m_a $$. Power saving calculations appear frequently (e.g., Jun 2025, Jun 2023).

3. Angle Modulation (FM & PM)

Fundamentals

• Angle modulation: $$\displaystyle s(t) = A_c \cos(\omega_c t + \phi(t)) $$, where $\phi(t)$ is phase deviation.

• FM: $$\displaystyle \phi(t) = K_f \int m(\tau) d\tau $$ → instantaneous frequency $$\displaystyle f_i = f_c + \frac{K_f}{2\pi} m(t) $$.

• PM: $$\displaystyle \phi(t) = K_p m(t) $$ → instantaneous frequency $$\displaystyle f_i = f_c + \frac{K_p}{2\pi} \frac{dm(t)}{dt} $$.

• For sinusoidal $$\displaystyle m(t) = A_m \cos(2\pi f_m t) $$:

  • FM: $$\displaystyle \phi(t) = \beta \sin(2\pi f_m t) $$, $$\displaystyle \beta = \frac{\Delta f}{f_m} $$, $$\displaystyle \Delta f = K_f A_m $$.

  • PM: $$\displaystyle \phi(t) = \beta_p \cos(2\pi f_m t) $$, $$\displaystyle \beta_p = K_p A_m $$.

Frequency Modulation (FM)

• Modulation index: $$\displaystyle \beta = \frac{\Delta f}{f_m} $$

• Carson's rule (bandwidth approximation):

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

• Narrowband FM (NBFM) ($\beta \ll 1$):

  $$\displaystyle s(t) \approx A_c \left[ \cos(\omega_c t) - \beta \sin(\omega_c t) \sin(\omega_m t) \right] $$

  Spectrum: Carrier + two sidebands (like AM but sidebands $$\displaystyle 180^\circ $$ out of phase). Bandwidth $$\displaystyle = 2f_m $$.

• Wideband FM ($\beta \gg 1$): Many sidebands, bandwidth given by Carson.

FM Generation Methods

• Direct method: VCO or reactance modulator. Instantaneous frequency directly proportional to $m(t)$.

• Indirect method (Armstrong): Phase modulator with integrated $m(t)$, followed by frequency multiplier to increase deviation.

FM Demodulation Methods

• Frequency discriminators:

  • Foster-Seeley: Balanced circuit with two tuned circuits (one leading, one lagging). Output proportional to frequency deviation.

  • Ratio detector: Limiter + detector; less sensitive to amplitude variations.

• Phase-Locked Loop (PLL) detector:

  • Block diagram: Phase comparator → Loop filter → VCO.

  • Operation: Input FM compared with VCO output; error voltage (after filtering) is demodulated message.

  • Advantages: Noise immunity, stability, capture effect.

  • Tracking range: Range over which PLL can lock; Hold-in range: Maximum frequency deviation for which lock is maintained.

Pre-emphasis & De-emphasis

• Need: FM demodulator output noise has high-frequency emphasis (differentiator effect). Pre-emphasis boosts high frequencies before transmission; de-emphasis attenuates them at receiver → improves SNR at high frequencies.

• Networks:

  • Pre-emphasis: High-pass RC (differentiator-like), time constant $$\displaystyle \tau = \frac{1}{2\pi f_c} $$.

  • De-emphasis: Low-pass RC (integrator-like), same $\tau$.

• SNR improvement: For sinusoidal modulation at $$\displaystyle f_m $$, improvement factor $$\displaystyle = 1 + \left(\frac{f_c}{f_m}\right)^2 $$.

  Derivation: Without pre-emphasis, output $$\displaystyle \text{SNR} \propto \frac{\beta^2}{f_m^2 N_0} $$. With pre-emphasis, signal power at $$\displaystyle f_m $$ multiplied by $$\displaystyle |H_{pre}(f_m)|^2 $$, noise power multiplied by $$\displaystyle |H_{de}(f_m)|^2 $$, giving factor $$\displaystyle 1 + (f_c/f_m)^2 $$.

[!TIP]

Exam Focus: Carson's rule is essential. For NBFM, write the approximate equation. Pre-emphasis/de-emphasis networks and SNR improvement derivation are common (e.g., Jun 2023, Jun 2025).

4. Noise in Analog Communication Systems

Noise Fundamentals

• Classification:

  • External: Atmospheric (lightning), galactic (cosmic), industrial (vehicles, appliances).

  • Internal: Thermal (Johnson-Nyquist), shot (semiconductors), transit-time (high frequency).

• Noise figure ($F$):

  $$\displaystyle F = \frac{\text{SNR}_\text{in}}{\text{SNR}_\text{out}} $$ (same bandwidth). In dB: $$\displaystyle NF = 10 \log_{10} F $$.

• Noise temperature ($$\displaystyle T_e $$):

  $$\displaystyle T_e = (F - 1) T_0 $$, where $$\displaystyle T_0 = 290\,\text{K} $$ (reference).

• Noise bandwidth ($$\displaystyle B_n $$):

  $$\displaystyle B_n = \frac{1}{|H(f_0)|^2} \int_{-\infty}^{\infty} |H(f)|^2 df $$; for rectangular filter, $$\displaystyle B_n = B $$.

• Friis formula (cascaded amplifiers):

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

Noise in Modulation Systems

• AM (envelope detector):

  • At high SNR: $$\displaystyle \text{SNR}_\text{out} \approx \frac{m_a^2 P_c}{2 N_0 B} $$

    $$\displaystyle \text{SNR}_\text{in} = \frac{P_c}{N_0 B} $$

    Figure of merit: $$\displaystyle \boxed{\frac{\text{SNR}_\text{out}}{\text{SNR}_\text{in}} = \frac{m_a^2}{2 + m_a^2}} $$

• DSB-SC (coherent):

  • $$\displaystyle \text{SNR}_\text{out} = \frac{P_m}{N_0 f_m} $$, $$\displaystyle \text{SNR}_\text{in} = \frac{P_m/2}{2 N_0 f_m} = \frac{P_m}{4 N_0 f_m} $$

    Figure of merit: $\boxed{4}$

• SSB-SC (coherent):

  • $$\displaystyle \text{SNR}_\text{out} = \frac{P_m}{N_0 f_m} $$, $$\displaystyle \text{SNR}_\text{in} = \frac{P_m}{N_0 f_m} $$

    Figure of merit: $\boxed{1}$

• FM:

  • Threshold effect: Below certain $\beta$, SNR degrades rapidly.

  • Capture effect: Stronger signal suppresses weaker ones.

  • Figure of merit (large $\beta$): $$\displaystyle \boxed{\frac{3}{2} \beta^2} $$

  • Pre-emphasis improves SNR by factor $$\displaystyle 1 + (f_c/f_m)^2 $$.

[!TIP]

Key Comparison: AM (FOM $$\displaystyle = m_a^2/(2+m_a^2) $$) vs. DSB-SC (FOM $$\displaystyle = 4 $$) vs. FM (FOM $$\displaystyle = \frac{3}{2}\beta^2 $$). FM provides greatest noise immunity for large $\beta$.

5. Receiver Architectures & Performance

TRF (Tuned Radio Frequency) Receiver

• Block diagram:

  DiagramCANVAS: RF amplifier → Tuned RF stages → Detector → AF amplifier → Speaker

• Operation: All amplification at RF; successive tuned circuits select desired station.

• Advantages: Simple, no image frequency.

• Limitations:

  • Instability (high Q at high frequencies).

  • Poor selectivity (difficult to achieve narrow bandwidth at high RF).

  • Adjacent channel interference.

  • Image frequency not an issue but poor rejection of nearby stations.

Superheterodyne Receiver

• Block diagram:

  DiagramCANVAS: RF amp → Mixer + LO → IF amp → Detector → AF amp → Speaker; with AGC loop

• Operation: Frequency conversion to fixed IF (e.g., 455 kHz for AM, 10.7 MHz for FM) for easier amplification and filtering.

• Intermediate Frequency (IF) Choice:

  • Lower IF → better selectivity (higher Q achievable).

  • Higher IF → better image rejection ($$\displaystyle f_\text{img} = f_\text{sig} + 2IF $$).

  • Must avoid integer multiples of IF in band.

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

• Image Frequency & Rejection:

  • Origin: Mixer produces sum and difference frequencies; signal at $$\displaystyle f_\text{img} = f_\text{sig} + 2IF $$ also produces IF.

  • Image rejection ratio (IFRR):

    $$\displaystyle \boxed{\text{IFRR} = 1 + \left( \frac{2Q f_\text{IF}}{f_\text{sig}} \right)^2} $$

    where $Q$ is Q of RF tuned circuit.

  • Improvement: Increase RF amplifier Q, use additional tuned stages.

• Selectivity: Ability to separate adjacent channels (determined by IF filter bandwidth).

• Fidelity: Accuracy of audio reproduction (determined by AF response).

Automatic Control Circuits

• AGC (Automatic Gain Control) / AVC:

  • Need: Maintain constant output despite signal strength variations.

  • Operation: Detector output filtered to generate DC voltage → controls gain of RF/IF amplifiers.

• AFC (Automatic Frequency Control):

  • Need: Stabilize local oscillator frequency.

  • Operation: Frequency discriminator provides DC error voltage → tunes LO (varactor diode).

[!TIP]

Exam Focus: Image frequency calculation ($$\displaystyle f_\text{img} = f_\text{sig} + 2IF $$) and IFRR derivation are common (e.g., Jun 2023, Jun 2022). Compare TRF vs. superheterodyne.

6. Specialized Modulation & Detection Techniques

SSB-SC Generation Details

• Phase discrimination method:

  • Generate Hilbert transform $$\displaystyle m_h(t) $$ of $m(t)$ (using $$\displaystyle 90^\circ $$ phase-shift network).

  • Form: $$\displaystyle s(t) = m(t) \cos(\omega_c t) \mp m_h(t) \sin(\omega_c t) $$.

  • Block diagram:

    DiagramCANVAS: Input m(t) → split → path1: × cos → path2: Hilbert transform → × (-sin) → sum → SSB-SC

• Balanced modulator + filter: Generate DSB-SC via balanced modulator, then filter out one sideband.

VSB-SC Applications

• TV broadcasting: Video signal has large low-frequency content; full SSB filtering near carrier is impractical. VSB preserves low frequencies while saving ~25% bandwidth compared to DSB.

• Generation: DSB-SC followed by vestigial filter (asymmetric shape).

• Detection: Coherent detection with carrier recovery; if carrier partially transmitted, envelope detection possible.

Costas Loop

• Purpose: Demodulate DSB-SC (carrier recovery).

• Block diagram:

  DiagramCANVAS: Input → split → path1: × cos(ω_c t) → LPF → V1; path2: × sin(ω_c t) → LPF → V2; V1 × cos - V2 × sin → output

• Operation: Two phase detectors (0° and 90°). Loop adjusts VCO phase to maximize output, recovering carrier. Output from product of V1 and V2 gives baseband.

FM Transmitter & Receiver

• FM transmitter block diagram:

  DiagramCANVAS: Pre-emphasis → Modulator (VCO) → Power amplifier → Antenna

• FM receiver block diagram:

  DiagramCANVAS: RF amp → Mixer → IF amp (limiter) → Discriminator/PLL → De-emphasis → AF amp → Speaker

[!TIP]

Short Note Topics: Costas Loop (7m), VSB-SC (4m), SSB generation methods (7m) are frequently asked.

7. Key Calculations & Problem-Solving Topics

Fourier Transform Problems

• Piecewise signals: Split into intervals, integrate $$\displaystyle X(f) = \int x(t) e^{-j2\pi f t} dt $$.

• Properties: Practice time scaling, duality proofs.

AM Calculations

• Modulation index from currents:

  $$\displaystyle I_t = I_c \sqrt{1 + \frac{m_a^2}{2}} \quad \Rightarrow \quad m_a = \sqrt{2\left[\left(\frac{I_t}{I_c}\right)^2 - 1\right]} $$

• Power:

  $$\displaystyle P_c = \frac{I_c^2 R}{2} $$, $$\displaystyle P_\text{sidebands} = \frac{m_a^2}{2} P_c $$, $$\displaystyle P_t = P_c\left(1 + \frac{m_a^2}{2}\right) $$

• % power saving: See Section 2.

FM Calculations

• Modulation index: $$\displaystyle \beta = \frac{\Delta f}{f_m} $$, $$\displaystyle \Delta f = K_f V_m $$

• Carson's rule: $$\displaystyle BW \approx 2(\Delta f + f_m) $$

• Spectrum: Sideband amplitudes: $$\displaystyle A_c J_n(\beta) $$ for $$\displaystyle n^{th} $$ sideband.

• Effect of changing modulating signal:

  $$\displaystyle \Delta f \propto V_m $$, $$\displaystyle \beta \propto \frac{V_m}{f_m} $$.

Receiver Calculations

• Image frequency: $$\displaystyle f_\text{img} = f_\text{sig} + 2 f_\text{IF} $$

• Image rejection ratio: $$\displaystyle \text{IFRR} = 1 + \left( \frac{2Q f_\text{IF}}{f_\text{sig}} \right)^2 $$

Noise Calculations

• Friis formula: $$\displaystyle F_\text{total} = F_1 + \frac{F_2-1}{G_1} + \frac{F_3-1}{G_1 G_2} + \cdots $$

• Figure of merit derivations: For AM, DSB-SC, FM (see Section 4).

8. Short Note Topics (Frequently Asked)

Pre-emphasis & De-emphasis

• Need: Compensate for high-frequency noise boost in FM demodulator.

• Pre-emphasis: High-pass RC network (boost high frequencies).

• De-emphasis: Low-pass RC network (attenuate high frequencies).

• SNR improvement: Factor $$\displaystyle 1 + (f_c/f_m)^2 $$ for sinusoidal $$\displaystyle f_m $$.

PLL Detector

• Block diagram: Phase comparator → Loop filter → VCO.

• Operation: Input FM compared with VCO output; error voltage (filtered) is demodulated message.

• Advantages: Noise immunity, stability, capture effect.

VSB-SC

• Need: Practical filtering for signals with low-frequency content (e.g., TV video).

• Generation: DSB-SC followed by vestigial filter (asymmetric).

• Detection: Coherent detection; if carrier present, envelope detection possible.

AGC (Automatic Gain Control)

• Need: Maintain constant output despite signal strength variations.

• Operation: Detector output filtered → DC control voltage → varies gain of RF/IF amplifiers.

FM Transmitter

• Blocks: Pre-emphasis → Modulator (VCO/reactance modulator) → Power amplifier.

• Modulation: Direct (VCO) or indirect (phase modulator + multiplier).

Convolution Theorem

• Time convolution ↔ Frequency multiplication: $$\displaystyle x(t)*h(t) \leftrightarrow X(f)H(f) $$.

• Frequency convolution ↔ Time multiplication: $$\displaystyle x(t)h(t) \leftrightarrow \frac{1}{2\pi} X(f)*H(f) $$.

IF Frequency

• Choice factors: Selectivity (lower IF better), image rejection (higher IF better), ease of amplification, adjacent channel interference.

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

Noise Bandwidth

• Definition: $$\displaystyle B_n = \frac{1}{|H(f_0)|^2} \int |H(f)|^2 df $$.

• For rectangular filter, $$\displaystyle B_n = B $$.

AFC (Automatic Frequency Control)

• Need: Stabilize local oscillator frequency.

• Operation: Frequency discriminator provides DC error voltage → tunes LO (varactor).

Selectivity

• Ability to separate adjacent channels.

• Determined by IF filter bandwidth and Q.

Image Signal Rejection

• Image frequency: $$\displaystyle f_\text{img} = f_\text{sig} + 2 f_\text{IF} $$.

• Rejection ratio: $$\displaystyle \text{IFRR} = 1 + \left( \frac{2Q f_\text{IF}}{f_\text{sig}} \right)^2 $$.

• Improvement: Increase RF amplifier Q, use additional tuned stages.

Costas Loop

• Used for DSB-SC demodulation and carrier recovery.

• Two phase detectors (0° and 90°); loop adjusts VCO phase.

• Output from product of detector outputs gives baseband.

Narrowband FM vs. AM

• NBFM equation: $$\displaystyle s(t) \approx A_c \left[ \cos(\omega_c t) - \beta \sin(\omega_c t) \sin(\omega_m t) \right] $$.

• Waveform comparison: NBFM has constant envelope (like FM), AM envelope varies with $m(t)$.

• Bandwidth: Both $$\displaystyle 2f_m $$ for NBFM and AM.

• Noise immunity: NBFM slightly better than AM at high SNR? Actually, NBFM offers no SNR improvement over AM (proved in Jun 2022).