Analog Communication (EC-403) - Unit 3 Short Notes

UNIT 3: ANALOG COMMUNICATION - EXAM-FOCUSED SHORT NOTES

1. SIGNAL ANALYSIS & FOURIER TRANSFORM (FOUNDATION)

Properties of Fourier Transform (FT)

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

  Significance: Shift in time domain introduces linear phase shift in frequency domain.

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

  Significance: Modulation in time domain translates spectrum in frequency.

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

  Significance: Compression in time expands spectrum and vice-versa.

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

• Convolution Theorem: $$\displaystyle x_1(t) * x_2(t) \leftrightarrow X_1(f)X_2(f) $$

  Significance: Simplifies analysis of LTI systems; multiplication in freq. domain ⇔ convolution in time.

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

  Significance: Total energy of signal is same in time and frequency domains.

[!TIP] Exam Focus: Proofs of Time/Frequency shifting and Convolution theorem are frequently asked. Be clear with the duality property for reverse problems.

Fourier Transform of Standard Signals

Correlation Functions & Spectral Density

• Auto-correlation: $$\displaystyle R_{xx}(\tau) = \int_{-\infty}^{\infty} x(t)x^*(t-\tau) dt $$ (energy) or $$\displaystyle R_{xx}(\tau) = \lim_{T\to\infty} \frac{1}{T}\int_{-T/2}^{T/2} x(t)x^*(t-\tau) dt $$ (power).

  • Properties: Even function, max at $$\displaystyle \tau=0 $$, $$\displaystyle R_{xx}(0) $$ = Energy/Power.

• Cross-correlation: $$\displaystyle R_{xy}(\tau) = \int_{-\infty}^{\infty} x(t)y^*(t-\tau) dt $$.

  • Application: Used for signal detection in noise, time delay estimation.

• Energy Spectral Density (ESD): $$\displaystyle E_x(f) = |X(f)|^2 $$. $$\displaystyle R_{xx}(\tau) \leftrightarrow E_x(f) $$.

• Power Spectral Density (PSD): $$\displaystyle S_x(f) = \lim_{T\to\infty} \frac{|X_T(f)|^2}{T} $$. $$\displaystyle R_{xx}(\tau) \leftrightarrow S_x(f) $$.

• Wiener-Khinchin Theorem: Auto-correlation function and Power Spectral Density form a Fourier Transform pair.

[!TIP] Common Pitfall: Remember ESD is for finite energy signals, PSD for power signals. The transform pair is between Auto-correlation and PSD (for WSS processes).

2. AMPLITUDE MODULATION (AM) SYSTEMS

Principle & Need for Modulation

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

• Key Needs:

  1. Antenna Size: $h \propto \lambda$; modulation shifts signal to higher freq for practical antenna.

  2. Frequency Translation: Enables multiplexing (FDM) and avoids signal overlap.

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

  4. Bandwidth Allocation: Regulatory assignment of specific frequency bands.

Standard 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 $$\displaystyle m_a = \frac{V_m}{V_c} $$ (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 $$\displaystyle f_c-f_m $$.

• Power Relations:

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

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

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

  • Transmission Efficiency: $$\displaystyle \eta = \frac{P_{SB}}{P_T} = \frac{m_a^2}{2+m_a^2} $$

• Over-modulation: $$\displaystyle m_a > 1 $$ causes envelope distortion, not recoverable by envelope detector.

Suppressed Carrier Techniques

• DSB-SC:

  • Generation: Balanced Modulator (multiplier), Ring Modulator.

  • Spectrum: Only USB and LSB, no carrier.

  • Demodulation: Synchronous/Coherent Detection (multiply by $$\displaystyle 2\cos(2\pi f_c t) $$ then LPF). Requires carrier phase sync.

• SSB-SC:

  • Need: 50% bandwidth & power saving over DSB-FC.

  • Generation Methods:

    1. Filter Method: Use sharp bandpass filter to remove one sideband from DSB-SC.

    2. Phase Discrimination (Hilbert Transform): Uses $$\displaystyle 90^\circ $$ phase-shifted versions of $m(t)$. $$\displaystyle s(t) = m(t)\cos(2\pi f_c t) \pm \hat{m}(t)\sin(2\pi f_c t) $$.

  • Demodulation: Synchronous detection with coherent carrier.

• VSB-SC:

  • Need: Compromise for TV (vestige of other sideband retained for easier filter design).

  • Filter Shape: One sideband passed almost completely, a vestige (tail) of the other sideband is also passed.

Modulator & Demodulator Circuits

• Square Law Modulator: Uses non-linear device $$\displaystyle i = a_0 + a_1 v + a_2 v^2 $$. For $$\displaystyle v = A_c\cos(2\pi f_c t) + m(t) $$, output contains $$\displaystyle A_c m(t)\cos(2\pi f_c t) $$ term (DSB-SC component). Low power efficiency.

• Envelope Detector: Diode + RC filter. Output follows envelope of AM signal.

  • Distortion Criteria: $$\displaystyle m_a \leq 1 $$ (no over-modulation). RC time constant: $$\displaystyle \frac{1}{f_c} \ll RC \ll \frac{1}{f_m} $$.

• Synchronous Detector: Multiplies received signal by a locally generated carrier $$\displaystyle \cos(2\pi f_c t + \phi) $$. Output after LPF is $$\displaystyle \frac{A_c m(t)}{2}\cos\phi $$. Requires phase lock ($\phi \approx 0$).

Comparisons & Power Saving

• % Power Saving when carrier & one sideband suppressed:

$$\% \text{Saving} = \left(1 - \frac{P_{remaining}}{P_T}\right) \times 100\%$$

For SSB-SC (only one sideband remains): $$\displaystyle P_{remaining} = \frac{m_a^2 P_c}{4} $$ (half of sideband power).  

For $$\displaystyle m_a=1 $$ (100%): $$\displaystyle \% \text{Saving} = \left(1 - \frac{1/4}{1.5}\right)\times100 = \boxed{83.33\%} $$  

For $$\displaystyle m_a=0.5 $$ (50%): $$\displaystyle \% \text{Saving} = \left(1 - \frac{0.25/4}{1+0.25/2}\right)\times100 = \boxed{94.12\%} $$

[!TIP] Exam Trap: Power saving calculation uses original total power $$\displaystyle P_T $$ of the full AM (DSB-FC) wave as reference, not the carrier power alone.

3. ANGLE MODULATION (FM & PM)

Fundamentals

• Instantaneous Phase: $$\displaystyle \theta_i(t) = 2\pi f_c t + \phi(t) $$

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

• FM vs. PM:

  • FM: $$\displaystyle \phi(t) = k_f \int m(\tau)d\tau \Rightarrow \beta_{FM} = \frac{k_f A_m}{f_m} $$ (Δf = $$\displaystyle k_f A_m $$)

  • PM: $$\displaystyle \phi(t) = k_p m(t) \Rightarrow \beta_{PM} = k_p A_m $$

  • Relation: For same $m(t)$, FM's $\beta$ is inversely proportional to $$\displaystyle f_m $$, PM's $\beta$ is independent of $$\displaystyle f_m $$.

• Narrowband FM (NBFM): $\beta \ll 1$. $$\displaystyle s(t) \approx A_c \cos(2\pi f_c t) - \frac{\beta A_c}{2}[m(t)\sin(2\pi f_c t) + \hat{m}(t)\cos(2\pi f_c t)] $$. Bandwidth $$\displaystyle \approx 2f_m $$.

• Wideband FM (WBFM): $\beta \gg 1$. Carson's Rule: $$\displaystyle BW \approx 2(\Delta f + f_m) = 2f_c(\beta + 1) $$ where $$\displaystyle \Delta f = \beta f_m $$ (max freq. deviation).

Generation of FM

• Direct Method (VCO/Reactance Modulator): $$\displaystyle v_{out} = V_c \cos(2\pi f_c t + k_f \int m(\tau)d\tau) $$. Reactance modulator (diode/transistor) presents a voltage-controlled reactance to an LC oscillator.

• Indirect Method (Armstrong): Phase modulator + frequency multiplier. First generate NBFM, then use frequency multipliers (non-linear devices) to increase $\Delta f$ and $$\displaystyle f_c $$ while keeping $\beta$ constant.

Demodulation of FM

• Frequency Discriminators:

  • Foster-Seeley Discriminator: Uses two tuned circuits (primary & secondary) at $$\displaystyle f_c $$, $$\displaystyle 90^\circ $$ apart. Output voltage proportional to input frequency deviation. Linear range centered at $$\displaystyle f_c $$.

  • Ratio Detector: Variation of Foster-Seeley with diode in series with tank. Advantage: No need for linear limiter before detector; suppresses AM noise.

• Phase-Locked Loop (PLL) Detector:

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

  • Operation: PD compares phase of input FM and VCO output. Error voltage (after LF) drives VCO. In lock, VCO control voltage is a replica of modulating signal $m(t)$.

  • Capture Range: Range of input frequencies over which PLL acquires lock.

  • Lock Range (Hold-in Range): Range over which PLL maintains lock once acquired.

Pre-emphasis & De-emphasis

• Need: FM has good SNR for high frequencies, but message signals often have more power at low frequencies (e.g., speech). Pre-emphasis boosts high frequencies at transmitter to equalize SNR.

• Networks:

  • Pre-emphasis: High-pass RC network (differentiator). Transfer function: $$\displaystyle H_{pre}(f) = 1 + j2\pi f \tau $$.

  • De-emphasis: Low-pass RC network (integrator). Transfer function: $$\displaystyle H_{de}(f) = \frac{1}{1 + j2\pi f \tau} $$.

  • $$\displaystyle \tau = RC = 1/(2\pi f_c) $$ where $$\displaystyle f_c $$ is corner freq (e.g., 2.1 kHz for US, 50 µs).

• SNR Improvement: High-frequency noise is attenuated by de-emphasis network, while pre-emphasis ensures signal power at high frequencies is increased. Net improvement $$\displaystyle \approx 10\log_{10}(1 + (f/f_c)^2) $$ dB.

Spectrum & Modulation Index

• FM Signal: $$\displaystyle s(t) = A_c \cos(2\pi f_c t + \beta \sin(2\pi f_m t)) $$

• Bessel Function Expansion: $$\displaystyle s(t) = A_c \sum_{n=-\infty}^{\infty} J_n(\beta) \cos(2\pi (f_c + n f_m) t) $$

  • Sideband at $$\displaystyle f_c \pm n f_m $$ has amplitude $$\displaystyle A_c J_n(\beta) $$.

  • Significant Sidebands: Those with $$\displaystyle |J_n(\beta)| > 0.01 $$ (or 0.1). Bandwidth $$\displaystyle \approx 2n_{max} f_m $$ where $$\displaystyle n_{max} $$ is largest $n$ with significant $$\displaystyle J_n $$.

• Given $$\displaystyle S(t)=A_c\cos(2\pi f_c t + \beta \sin 2\pi f_m t) $$:

  • Frequency Deviation: $$\displaystyle \Delta f = \beta f_m $$

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

[!EXAMPLE] Problem Solving:

For $$\displaystyle S(t)=10\cos\left(2\pi \times 10^8 t + 5\sin 8\pi t\right) $$:

Compare with $$\displaystyle S(t)=A_c\cos(2\pi f_c t + \beta \sin 2\pi f_m t) $$.

$$\displaystyle 2\pi f_m t = 8\pi t \Rightarrow f_m = 4\,\text{Hz} $$.

$$\displaystyle \beta = 5 $$.

$$\displaystyle \Delta f = \beta f_m = 5 \times 4 = 20\,\text{Hz} $$.

$$\displaystyle f_c = 10^8\,\text{Hz} $$.

Carson's BW: $$\displaystyle BW \approx 2(\Delta f + f_m) = 2(20+4) = 48\,\text{Hz} $$.

4. NOISE IN ANALOG COMMUNICATION SYSTEMS

Noise Fundamentals

• Classification:

  • External: Atmospheric, Extraterrestrial (cosmic), Industrial.

  • Internal: Thermal (Johnson-Nyquist), Shot, Flicker ($1/f$), Transit-time.

• Noise Figure (NF): $$\displaystyle NF = \frac{(SNR)_{in}}{(SNR)_{out}} $$ (linear) or $$\displaystyle NF(dB) = 10\log_{10}(NF) $$. Measures degradation in SNR by a device.

• Noise Temperature ($$\displaystyle T_e $$): $$\displaystyle T_e = T_0 (NF - 1) $$ where $$\displaystyle T_0 = 290\,\text{K} $$ (standard). Represents extra noise contributed by device.

• Noise Bandwidth ($$\displaystyle B_N $$): Equivalent bandwidth of an ideal filter that passes same noise power as actual filter. For RC LPF: $$\displaystyle B_N = \frac{1}{4RC} $$.

• Measurement (Y-factor): $$\displaystyle Y = \frac{P_{out,on}}{P_{out,off}} = 1 + \frac{NF T_s}{T_0} $$. $$\displaystyle NF = \frac{Y-1}{T_s/T_0} $$ where $$\displaystyle T_s $$ is noise source temp.

Noise in AM Receivers

• Envelope Detector: Output SNR $$\displaystyle = \frac{m_a^2 P_c}{2N_0 B} $$ (for large SNR). Degrades rapidly as $$\displaystyle m_a \to 0 $$.

• Synchronous Detector: Output SNR $$\displaystyle = \frac{m_a^2 P_c}{4N_0 B} $$ (worse by 3 dB? Actually, careful: For DSB-SC coherent, SNR_out = (m_a^2 P_c)/(4N_0 B). For AM with carrier, coherent gives same as DSB-SC but with carrier power present? Standard result: Coherent AM has same SNR as DSB-SC. Envelope detector has $$\displaystyle \frac{m_a^2}{2+m_a^2} $$ factor).

• Figure of Merit for AM: $$\displaystyle G_{AM} = \frac{(SNR_o/SNR_i)_{AM}}{(SNR_o/SNR_i)_{DSB-SC}} = \frac{m_a^2}{2+m_a^2} $$ (for envelope detector). Max $$\displaystyle G_{AM}=0.5 $$ (3 dB) at $$\displaystyle m_a=1 $$.

Noise in FM Receivers

• Threshold Effect: For low SNR at FM detector input, output SNR drops drastically (non-linear effect). Threshold $$\displaystyle SNR_{in} \approx 10-16\,\text{dB} $$.

• Figure of Merit for FM: $$\displaystyle G_{FM} = \frac{3}{2} \beta^2 $$ (for large $\beta$, pre-emphasis/de-emphasis ignored). Shows quadratic improvement with $\beta$.

• Comparison: FM offers much higher $$\displaystyle G_{FM} $$ than AM's $$\displaystyle G_{AM} $$ for large $\beta$. Reason: FM demodulator is insensitive to amplitude noise.

[!TIP] Key Insight: FM's noise immunity comes from its constant envelope; noise only affects amplitude, which FM detector ignores. AM's information is in amplitude, so directly corrupted.

5. RECEIVERS & PERFORMANCE PARAMETERS

Receiver Types

• TRF (Tuned Radio Frequency) Receiver:

  • Block: RF Amp → Mixer (not present, direct) → Multiple TRF stages → Detector → Audio Amp.

  • Advantages: Simple, no image freq.

  • Limitations: Instability (many tuned stages at high freq), Poor selectivity (hard to make sharp filters at high freq), Image freq. problem (if any mixing occurs), Poor adjacent channel rejection.

• Superheterodyne Receiver:

  • Block Diagram: RF Amplifier → Mixer (with LO) → IF Amplifier (fixed freq) → Detector → Audio Amplifier

  • Function of Blocks:

    1. RF Amp: Selects desired signal, provides initial gain, rejects image.

    2. Mixer: Heterodynes RF with LO to produce fixed IF ($$\displaystyle f_{IF} = |f_{RF} - f_{LO}| $$).

    3. IF Amp: Provides most gain and selectivity (sharp filters at fixed $$\displaystyle f_{IF} $$).

    4. Detector: Demodulates IF signal.

    5. Audio Amp: Amplifies baseband.

  • Additional Circuits: AGC (controls gain), AFC (controls LO freq).

Critical Parameters in Superheterodyne

• Intermediate Frequency (IF): Choice factors:

  1. Image Rejection: Lower IF → Easier image rejection (better IFRR).

  2. Selectivity: Lower IF → Easier to achieve sharp filters.

  3. Stability: Higher IF → Less prone to LO radiation & image problems.

  4. Adjacent Channel Interference: Higher IF → Better rejection.

  • Common Choice: 455 kHz (AM), 10.7 MHz (FM).

• Image Frequency ($$\displaystyle f_{img} $$):

  • Definition: Unwanted frequency that also produces output at $$\displaystyle f_{IF} $$.

  • Origin: $$\displaystyle f_{img} = f_{LO} + f_{IF} $$ (if $$\displaystyle f_{LO} > f_{RF} $$). A signal at $$\displaystyle f_{img} $$ mixes with LO to produce same IF as desired signal.

  • Image Frequency Rejection Ratio (IFRR): Ratio of output due to desired signal to output due to image signal.

$$\text{IFRR} = \frac{|H(f_{RF})|^2}{|H(f_{img})|^2}$$

    where $H(f)$ is transfer function of RF amp+preselector.

*   **Calculation:** For tuned RF amp with Q: $$\displaystyle f_{img} = f_c + 2f_{IF} $$ (if $$\displaystyle f_{LO} = f_c + f_{IF} $$). IFRR $$\displaystyle \approx Q \cdot \frac{f_c}{f_{IF}} $$ (approx).

• Selectivity: Ability to separate adjacent channels. Determined by IF filter bandwidth.

• Fidelity: Ability to reproduce original message accurately. Affected by linearity, phase distortion in IF filters, and detector characteristics.

Automatic Control Circuits

• AGC (Automatic Gain Control) / AVC:

  • Need: Compensate for varying signal strength (fading), maintain constant audio output.

  • Principle: Detects average envelope level, generates DC voltage to reduce RF/IF gain.

  • Effect: Improves receiver performance for weak/strong signals, but can cause distortion if AGC time constant is too short (chatter) or too long (slow response).

• AFC (Automatic Frequency Control):

  • Need: Correct small frequency drifts in LO (due to temp, aging).

  • Principle: Detector output (after LPF) contains DC proportional to freq. error. This voltage tunes LO capacitance (varactor) to correct drift.

  • Application: Stabilizes receiver tuning.

[!EXAMPLE] Image Rejection Problem:

Superhet tuned to $$\displaystyle f_c = 600\,\text{kHz} $$, $$\displaystyle f_{IF}=455\,\text{kHz} $$, RF amp Q=60.

$$\displaystyle f_{LO} = f_c + f_{IF} = 1055\,\text{kHz} $$.

$$\displaystyle f_{img} = f_{LO} + f_{IF} = 1055 + 455 = 1510\,\text{kHz} $$.

IFRR $$\displaystyle \approx Q \cdot \frac{f_c}{f_{IF}} = 60 \times \frac{600}{455} \approx 79.1 $$ (or $38.9\,\text{dB}$).

6. SPECIAL TOPICS & REVISION (Frequently Asked Short Notes)

PLL as FM Detector

• Block: PD → LF → VCO with feedback.

• Operation: Input FM signal has varying frequency. VCO tries to track it. Error voltage from PD (after LF) is proportional to frequency deviation → output $m(t)$.

• Advantages: High gain, good linearity, capture & lock range define performance.

VSB-SC Generation & Detection

• Generation: Filter DSB-SC with a filter having one sideband's cutoff at $$\displaystyle f_c $$ and the other's at $$\displaystyle f_c + f_v $$ (vestige $$\displaystyle f_v $$).

• Detection: Use coherent carrier (like DSB-SC/SSB) because VSB still has suppressed carrier. Alternatively, envelope detection possible if carrier is reinserted (like AM).

Convolution Theorem

• Statement: FT of convolution of two signals equals product of their individual FTs.

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

• Significance: Fundamental to LTI system analysis. Output spectrum = Input spectrum × System transfer function.

Noise Temperature vs. Noise Figure

Costas Loop for DSB-SC/SSB Demodulation

• Diagram: Two PDs (in-phase & quadrature), LF, VCO. VCO output split into $\cos$ and $-\sin$.

• Operation: PD1 (cos×cos) gives $m(t)\cos\phi + \text{noise}$, PD2 (cos×(-sin)) gives $m(t)\sin\phi + \text{noise}$. LF outputs control VCO to minimize $\phi \to 0$, then PD1 output is $m(t)$.

FM Transmitter Block Diagram

• Direct Method: Modulating Signal → Reactance Modulator → VCO (with carrier) → Power Amp → Antenna

• Indirect Method: Modulating Signal → Phase Modulator → Frequency Multiplier Chain → Power Amp → Antenna

Foster-Seeley Discriminator

• Circuit: Primary tuned to $$\displaystyle f_c $$, secondary tuned to $$\displaystyle f_c $$ with $$\displaystyle 90^\circ $$ phase shift (via capacitor). Diodes in series with secondary.

• Operation: At $$\displaystyle f_c $$, voltages across diodes are equal & opposite → zero output. Deviation causes imbalance → output voltage proportional to $\Delta f$.

• Linear Range: Symmetrical about $$\displaystyle f_c $$.

Balanced Modulator for DSB-SC

• Circuit: Two diodes (or transistors) in push-pull, driven by carrier (180° out of phase) and modulating signal.

• Operation: Carrier suppresses due to balanced connection. Output contains only cross-term $$\displaystyle m(t)\cos(2\pi f_c t) $$ (DSB-SC). Requires careful balance for good carrier suppression.

[!FINAL REMINDER] Exam Strategy:

1. Fourier Transforms: Memorize table. Practice shifting/scaling problems.

2. AM Power: Always calculate total power $$\displaystyle P_T = P_c(1+m_a^2/2) $$ first.

3. FM: Identify $\beta$ and $\Delta f$ from given $S(t)$. Apply Carson's rule.

4. Receivers: Draw superhet block diagram and label each block's function. Derive image freq.

5. Noise: Know $$\displaystyle G_{AM} = m_a^2/(2+m_a^2) $$ and $$\displaystyle G_{FM} \approx \frac{3}{2}\beta^2 $$. Explain threshold effect.

6. Comparisons: AM vs DSB-SC vs SSB-SC (BW, power, complexity). NBFM vs WBFM.