Analog Communication (EC-403) - Unit 5 Short Notes

I. Fourier Transform & Signal Analysis

A. Properties of Fourier Transform

Let \(X(f) = \mathcal{F}\{x(t)\}\). Key properties:

[!TIP]

Exam Focus: Time scaling often confused: \(x(at)\) compresses if \(|a|>1\), and FT magnitude scales by \(1/|a|\). Duality is frequently tested—swap time/frequency with sign flip and \(2\pi\) factor.

B. Fourier Transform of Basic Signals

[!TIP]

Common Pitfall: FT of \(u(t)\) includes an impulse at DC. Gate function FT is \(\tau \text{sinc}(f\tau)\)—remember \(\text{sinc}(x) = \sin(\pi x)/(\pi x)\).

C. Convolution Theorem

Statement:

Convolution in time domain ↔ Multiplication in frequency domain.

If \(y(t) = x_1(t) * x_2(t)\), then \(Y(f) = X_1(f) X_2(f)\).

Proof Sketch:

\[ Y(f) = \int_{-\infty}^{\infty} y(t) e^{-j2\pi f t} dt = \int_{-\infty}^{\infty} \left[ \int_{-\infty}^{\infty} x_1(\tau) x_2(t-\tau) d\tau \right] e^{-j2\pi f t} dt \]

Swap integrals and substitute \(u = t-\tau\):

\[ = \int_{-\infty}^{\infty} x_1(\tau) e^{-j2\pi f \tau} d\tau \cdot \int_{-\infty}^{\infty} x_2(u) e^{-j2\pi f u} du = X_1(f) X_2(f) \]

Applications:

• Simplifying LTI system analysis: output FT = input FT × system FT.

• Computing inverse FT of products via convolution.

D. Correlation

Auto-correlation:

\[ R_{xx}(\tau) = \int_{-\infty}^{\infty} x(t) x^*(t-\tau) dt \]

• Even function: \(R_{xx}(-\tau) = R_{xx}^*(\tau)\).

• Maximum at \(\tau=0\): \(R_{xx}(0) = E_x\) (total energy for energy signals, average power for power signals).

• Relation to Energy Spectral Density (ESD):

  \[ \psi_{xx}(f) = |X(f)|^2 \quad \text{and} \quad R_{xx}(\tau) = \int_{-\infty}^{\infty} \psi_{xx}(f) e^{j2\pi f \tau} df \]

  (Wiener-Khinchin for energy signals).

Cross-correlation:

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

• Not necessarily even.

• Relation to Cross-Spectral Density:

  \[ \psi_{xy}(f) = X(f) Y^*(f) \quad \text{and} \quad R_{xy}(\tau) = \int_{-\infty}^{\infty} \psi_{xy}(f) e^{j2\pi f \tau} df \]

• For power signals, use time-averaged correlation and Power Spectral Density (PSD).

E. Power Spectral Density (PSD)

Definition:

For a wide-sense stationary (WSS) random process \(x(t)\), PSD \(S_{xx}(f)\) is the Fourier transform of its auto-correlation function \(R_{xx}(\tau)\):

\[ S_{xx}(f) = \int_{-\infty}^{\infty} R_{xx}(\tau) e^{-j2\pi f \tau} d\tau \]

\[ R_{xx}(\tau) = \int_{-\infty}^{\infty} S_{xx}(f) e^{j2\pi f \tau} df \]

Properties:

• \(S_{xx}(f) \geq 0\) (non-negative).

• Even function: \(S_{xx}(-f) = S_{xx}(f)\).

• Total power: \(P = \int_{-\infty}^{\infty} S_{xx}(f) df = R_{xx}(0)\).

Wiener-Khinchin Theorem:

Auto-correlation and PSD are Fourier transform pairs.

II. Amplitude Modulation (AM)

A. Principles & Time-Domain Expression

Need for Modulation:

• Efficient antenna size (high frequency).

• Multiplexing (channel allocation).

• Reduced noise and interference.

• Flexibility in transmission.

Standard AM (DSB-TC):

\[ s(t) = [A_c + m(t)] \cos \omega_c t \]

where \(A_c\) = carrier amplitude, \(m(t)\) = message signal with \(|m(t)|_{\text{max}} \leq A_c\).

B. Spectrum & Modulation Index

Spectrum:

\[ S(f) = \frac{A_c}{2} [\delta(f-f_c) + \delta(f+f_c)] + \frac{1}{2} [M(f-f_c) + M(f+f_c)] \]

• Carrier at \(\pm f_c\).

• Upper Sideband (USB): \(M(f-f_c)\).

• Lower Sideband (LSB): \(M(f+f_c)\).

Modulation Index \(m\):

\[ m = \frac{|m(t)|_{\text{max}}}{A_c}, \quad 0 \leq m \leq 1 \text{ (no overmodulation)} \]
Bandwidth: \(BW = 2 f_m\), where \(f_m\) = max frequency in \(m(t)\).

C. Power Analysis & Power Saving

For single-tone \(m(t) = A_m \cos \omega_m t\):

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

• Sideband Power (each): \(P_{sb} = \frac{m^2 A_c^2}{4R}\)

• Total Power: \(P_t = P_c \left(1 + \frac{m^2}{2}\right)\)

• Power Efficiency: \(\eta = \frac{P_{sb,\text{total}}}{P_t} = \frac{m^2}{2 + m^2}\)

% Power Saving in Suppressed Carrier Systems:

• DSB-SC: Carrier suppressed → saving = \(\frac{P_c}{P_t} \times 100 = \frac{1}{1 + m^2/2} \times 100\)

• SSB-SC: Carrier + one sideband suppressed → saving = \(\frac{P_c + P_{sb}}{P_t} \times 100 = \frac{1 + m^2/2}{1 + m^2/2} \times 100 = 100\%\)? Wait: Actually for SSB-SC, only one sideband transmitted, so power = \(P_{sb}\). Compared to AM: saving = \(\frac{P_c + P_{other\,sb}}{P_t} \times 100 = \left(1 - \frac{P_{sb}}{P_t}\right) \times 100 = \left(1 - \frac{m^2/2}{1 + m^2/2}\right) \times 100 = \frac{1}{1 + m^2/2} \times 100\)? That's same as DSB-SC? No: In DSB-SC, both sidebands transmitted, no carrier: power = \(2 P_{sb} = \frac{m^2 A_c^2}{2R}\). AM total power = \(P_c + 2P_{sb}\). So DSB-SC saving vs AM: \(\frac{P_c}{P_t} \times 100 = \frac{1}{1 + m^2/2} \times 100\). SSB-SC: only one sideband, power = \(P_{sb}\). Saving vs AM: \(\frac{P_c + P_{sb}}{P_t} \times 100 = \frac{1 + m^2/2}{1 + m^2/2} \times 100?\) Actually:

\(P_t^{\text{AM}} = P_c + 2P_{sb}\)

\(P^{\text{SSB}} = P_{sb}\)

Saving = \(\frac{P_c + P_{sb}}{P_t^{\text{AM}}} \times 100 = \frac{P_c + P_{sb}}{P_c + 2P_{sb}} \times 100 = \frac{1 + m^2/2}{1 + m^2} \times 100\)? Let's compute:

\(P_c = A_c^2/(2R)\), \(P_{sb} = m^2 A_c^2/(4R)\)

So \(P_c + P_{sb} = \frac{A_c^2}{2R} (1 + m^2/2)\)

\(P_t^{\text{AM}} = \frac{A_c^2}{2R} (1 + m^2/2)\)? Wait: \(P_t^{\text{AM}} = P_c + 2P_{sb} = \frac{A_c^2}{2R} + 2 \cdot \frac{m^2 A_c^2}{4R} = \frac{A_c^2}{2R} (1 + m^2/2)\). So \(P_c + P_{sb} = P_t^{\text{AM}} - P_{sb}\). So saving = \(\frac{P_t^{\text{AM}} - P_{sb}}{P_t^{\text{AM}}} \times 100 = \left(1 - \frac{P_{sb}}{P_t^{\text{AM}}}\right) \times 100 = \left(1 - \frac{m^2/2}{1 + m^2/2}\right) \times 100 = \frac{1}{1 + m^2/2} \times 100\). That's same as DSB-SC? But DSB-SC transmits both sidebands, so its power is \(2P_{sb}\), saving vs AM: \(\frac{P_c}{P_t^{\text{AM}}} \times 100 = \frac{1}{1 + m^2/2} \times 100\). So both DSB-SC and SSB-SC have same % saving? That can't be right. Actually SSB-SC saves more because it also removes one sideband. So saving should be larger. Let's recalc:

AM total power: \(P_t = P_c + P_{USB} + P_{LSB} = P_c + 2P_{sb}\) (since symmetric).

DSB-SC: no carrier, both sidebands → \(P_{DSB} = 2P_{sb}\).

SSB-SC: no carrier, one sideband → \(P_{SSB} = P_{sb}\).

Saving vs AM:

• DSB-SC: \(\frac{P_c}{P_t} \times 100 = \frac{P_c}{P_c+2P_{sb}} \times 100 = \frac{1}{1 + 2P_{sb}/P_c} \times 100 = \frac{1}{1 + m^2/2} \times 100\)? Since \(P_{sb}/P_c = m^2/2\).

• SSB-SC: \(\frac{P_c + P_{sb}}{P_t} \times 100 = \frac{P_c + P_{sb}}{P_c+2P_{sb}} \times 100 = \frac{1 + P_{sb}/P_c}{1 + 2P_{sb}/P_c} \times 100 = \frac{1 + m^2/2}{1 + m^2} \times 100\).

For \(m=1\): DSB-SC saving = \(1/(1+0.5) \times 100 = 66.67\%\); SSB-SC saving = \((1+0.5)/(1+1) \times 100 = 1.5/2 \times 100 = 75\%\). Yes, SSB-SC saves more. So formula:

\[ \text{% Saving (SSB-SC)} = \frac{1 + \frac{m^2}{2}}{1 + m^2} \times 100 \]

D. Generation Techniques

1. Square Law Modulator:

   • Uses nonlinear device (diode): \(i = a_1 v + a_2 v^2\).

   • Input: \(v = A_c \cos \omega_c t + m(t)\).

   • Squaring term generates \(A_c m(t) \cos \omega_c t\) (DSB-SC) and carrier/sideband terms.

   • Filter out carrier → DSB-SC.

2. Balanced Modulator:

   • Two nonlinear devices in push-pull, carrier fed in opposite phase.

   • Carrier cancels at output → DSB-SC directly.

   • DiagramSEARCH: balanced modulator circuit diagram

3. Phase Discrimination Method (SSB-SC):

   • Filter Method: Generate DSB-SC, then filter out one sideband (sharp cutoff filter).

   • Phasing Method (Hilbert Transform):

     \[ s_{\text{SSB}}(t) = \frac{A_c}{2} m(t) \cos \omega_c t \mp \frac{A_c}{2} \hat{m}(t) \sin \omega_c t \]

     where \(\hat{m}(t)\) is Hilbert transform (90° phase shift).

   • DiagramSEARCH: phasing method SSB generator block diagram

E. Demodulation Techniques

• Envelope Detector:

  Diode + RC filter. Works if \(m \leq 1\) (no overmodulation). Output \(\propto |A_c + m(t)|\).

• Synchronous Detector (Coherent):

  Multiply received signal by locally generated \(\cos \omega_c t\) (phase-locked), then LPF.

  Works for DSB-SC and SSB-SC (with proper carrier phase).

• Costas Loop (for DSB-SC):

  Uses two phase-sensitive detectors (0° and 90° references) to recover carrier and data simultaneously.

  DiagramSEARCH: Costas loop block diagram

F. Comparison of AM, DSB-SC, SSB-SC, VSB-SC

G. Vestigial Sideband (VSB-SC)

Need:

• TV video: baseband 0–4 MHz (video) + audio at 4.5 MHz.

• SSB requires ideal filters (impossible). VSB transmits one full sideband + partial other (vestige) to ease filtering.

Generation:

1. Filter Method: Generate DSB-SC, pass through vestigial filter (passes one sideband fully, partial other).

2. Phasing Method: Similar to SSB but with imperfect 90° shift.

Detection:

• Use coherent detection (carrier synchronization) because VSB still has carrier.

• If carrier is transmitted (VSB-TC), envelope detection possible if modulation index small.

III. Angle Modulation (FM & PM)

A. Fundamentals & Equations

• FM:

  \[ s_{\text{FM}}(t) = A_c \cos\left[\omega_c t + k_f \int_{-\infty}^{t} m(\tau) d\tau\right] \]

  Instantaneous frequency: \(\omega_i(t) = \omega_c + k_f m(t)\).

• PM:

  \[ s_{\text{PM}}(t) = A_c \cos[\omega_c t + k_p m(t)] \]

  Instantaneous phase: \(\phi_i(t) = \omega_c t + k_p m(t)\).

Relationship:

FM is integral of PM. If \(m(t)\) is a sinusoid, both produce same waveform but with different interpretation of modulation index.

B. Narrowband FM (NBFM) vs Wideband FM (WBFM)

• NBFM (\(\beta \ll 1\)):

  \[ s(t) \approx A_c \cos \omega_c t - A_c \beta \sin \omega_c t \cdot m(t) \]

  Spectrum: carrier + two sidebands at \(f_c \pm f_m\), similar to DSB-SC but with carrier.

  Bandwidth \(\approx 2f_m\).

• WBFM (\(\beta \gg 1\)):

  Many sidebands (Bessel functions). Bandwidth given by Carson’s Rule.

C. Modulation Index, Frequency Deviation & Bandwidth

• Modulation Index \(\beta\):

  For sinusoidal \(m(t) = A_m \sin 2\pi f_m t\):

  \[ \beta = \frac{k_f A_m}{f_m} = \frac{\Delta f}{f_m} \]

  where \(\Delta f = k_f A_m\) = peak frequency deviation.

• Carson’s Rule:

  \[ \boxed{BW = 2(\Delta f + f_m)} \]

  Approximate bandwidth containing ~98% power.

Example:

Given \(s(t) = 10 \cos(2\pi f_c t + 5 \sin 8\pi t)\):

• Modulating frequency: \(8\pi t = 2\pi \cdot 4 \cdot t \Rightarrow f_m = 4 \text{ Hz}\)

• Peak phase deviation = 5 rad → \(\beta = 5\) (since for FM, \(\beta\) = peak phase deviation).

• \(\Delta f = \beta f_m = 5 \times 4 = 20 \text{ Hz}\)

• \(BW = 2(20 + 4) = 48 \text{ Hz}\).

D. Generation Methods

1. Direct Method:

   • Varactor Diode Modulator: Varactor capacitance varies with modulating voltage, placed in oscillator tank circuit.

   • Reactance Modulator: Transistor reactance converter.

2. Indirect Method (Armstrong):

   • Phase modulator (NBFM) + frequency multipliers (×2, ×3, ...) to increase \(\beta\) and \(f_c\).

   • Advantage: stable carrier frequency.

E. Demodulation Methods

1. Phase-Locked Loop (PLL) Detector:

   • PLL tracks FM signal; error voltage (after LPF) is demodulated output.

   • Wide capture range, good SNR.

2. Foster-Seeley Discriminator:

   • Two tuned RF circuits (primary & secondary) with 90° phase shift.

   • Frequency changes → amplitude variations at output.

   • DiagramSEARCH: Foster-Seeley discriminator circuit

3. Balanced Frequency Discriminator:

   • Variation of Foster-Seeley with center tap.

4. Phase Shift Discriminator:

   • RC network creates 90° phase shift at \(f_c\).

F. Pre-emphasis & De-emphasis

Need:

• FM noise power spectral density increases with frequency (\( \propto f^2 \)).

• High-frequency components of message signal more degraded.

Solution:

• Pre-emphasis (Transmitter): Boost high-frequency components using high-pass filter (differentiator).

  Transfer: \(H_{\text{pre}}(f) = 1 + j\frac{f}{f_c}\) (first-order).

• De-emphasis (Receiver): Attenuate high frequencies using low-pass filter (integrator).

  Transfer: \(H_{\text{de}}(f) = \frac{1}{1 + j\frac{f}{f_c}}\).

SNR Improvement:

Net signal transfer: \(H_{\text{pre}} H_{\text{de}} = 1\) (flat).

Noise: Pre-emphasis amplifies noise at high \(f\), but FM demodulator output noise \(\propto f^2\), and de-emphasis attenuates by \(1/\sqrt{1+(f/f_c)^2}\).

Overall, output SNR at frequency \(f\) improved by factor:

\[ \text{Improvement} = 1 + \left(\frac{f}{f_c}\right)^2 \]

For \(f \gg f_c\), improvement \(\approx (f/f_c)^2\).

IV. Noise in Analog Communication

A. Sources & Classification

B. Noise Parameters

• Noise Figure \(F\):

  \[ F = \frac{\text{SNR}_{\text{in}}}{\text{SNR}_{\text{out}}} \quad (\text{at same bandwidth}) \]

  \(F \geq 1\); in dB: \(F_{\text{dB}} = 10 \log_{10} F\).

• Noise Temperature \(T_e\):

  \[ T_e = (F - 1) T_0 \]

  where \(T_0 = 290 \text{ K}\) (standard).

• Noise Bandwidth \(B_N\):

  Equivalent rectangular bandwidth of filter that passes same noise power as actual filter.

C. Noise in AM Systems

• Envelope Detector:

  • Threshold effect: when carrier-to-noise ratio (CNR) drops below ~10 dB, output SNR degrades rapidly.

  • Output SNR for single-tone AM:

    \[ \left(\frac{S}{N}\right)_{\text{out}} = \frac{m^2 A_c^2}{8 R N_0 B} = \frac{m^2}{2} \cdot \frac{P_c}{N_0 B} \]

    where \(N_0\) = noise PSD, \(B\) = baseband bandwidth.

D. Noise in FM Systems

• Threshold Effect:

  When input CNR falls below threshold (~10–15 dB), demodulator loses lock, output SNR plummets.

• Capture Effect:

  FM receiver captures stronger signal, rejects weaker (within capture range).

• Output SNR (Wideband FM):

  \[ \left(\frac{S}{N}\right)_{\text{out}} \approx \frac{3}{2} \beta^2 \left(\frac{S}{N}\right)_{\text{IF}} \quad \text{for large } \beta \]

  Improves with \(\beta^2\).

E. Figure of Merit

Definition:

\[ F = \frac{(S/N)_{\text{out, baseband}}}{(S/N)_{\text{in, IF}}} \]

Derivation for AM (Envelope Detection):

\[ F_{\text{AM}} = \frac{m^2}{2 + m^2} \]

Derivation for DSB-SC (Coherent Detection):

\[ F_{\text{DSB-SC}} = \frac{m^2}{2} \]

(Assumes perfect carrier synchronization).

[!TIP]

Key Insight: DSB-SC has better figure of merit than AM because no carrier power wasted. For same modulation index, DSB-SC \(F\) is \(\frac{m^2}{2}\) vs AM \(\frac{m^2}{2+m^2}\).

F. Comparative Noise Performance

• FM vs AM:

  FM has superior noise immunity because:

  1. Limiter removes amplitude noise.

  2. Output SNR \(\propto \beta^2\) (can be very high).

  3. Capture effect rejects co-channel interference.

• Threshold Extension: FM systems use phase-locked demodulators to lower threshold effect.

V. Receivers

A. Tuned Radio Frequency (TRF) Receiver

Block Diagram:

RF amplifier → Mixer (not always) → Detector → Audio amplifier.

All stages tuned to carrier frequency.

Advantages:

• Simple, no image frequency.

Disadvantages:

• Instability (many tuned circuits).

• Poor selectivity (Q limited).

• Image frequency problem: \(f_{\text{image}} = f_{\text{signal}} + 2 f_{\text{IF}}\) (if IF used) or direct tuning issues.

B. Superheterodyne Receiver

Block Diagram & Function:

1. RF Amplifier: Selects desired channel, amplifies, provides initial image rejection.

2. Mixer: Heterodynes with LO \(f_{\text{LO}}\) to produce IF \(f_{\text{IF}} = |f_{\text{signal}} - f_{\text{LO}}|\).

3. Local Oscillator: \(f_{\text{LO}} = f_{\text{signal}} \pm f_{\text{IF}}\).

4. IF Amplifier: Fixed frequency (e.g., 455 kHz AM, 10.7 MHz FM), high gain, excellent selectivity.

5. Detector: Demodulates IF signal.

6. Audio Amplifier: Drives speaker.

Intermediate Frequency (IF):

• Choice Trade-off:

  • Higher IF → better image rejection (since \(f_{\text{image}} - f_{\text{signal}} = 2 f_{\text{IF}}\)).

  • Lower IF → better selectivity (since \(Q = f_{\text{IF}} / BW_{\text{IF}}\)).

• Typical: AM → 455 kHz; FM → 10.7 MHz.

Image Frequency:

\[ f_{\text{image}} = f_{\text{signal}} + 2 f_{\text{IF}} \quad (\text{if } f_{\text{LO}} = f_{\text{signal}} + f_{\text{IF}}) \]

Arises because mixer responds to both \(f_{\text{signal}}\) and \(f_{\text{image}}\) producing same IF.

Image Rejection Ratio (IMRR):

\[ \text{IMRR} = \frac{V_{\text{signal}}}{V_{\text{image}}} \approx Q \cdot \frac{f_{\text{signal}}}{f_{\text{IF}}} \]

where \(Q\) = Q-factor of RF tuned circuit.

Selectivity: Ability to separate adjacent channels (determined by IF filter shape).
Fidelity: Accuracy of audio reproduction.
Automatic Volume Control (AVC): DC from detector controls RF/IF gain to maintain constant output level.

C. Automatic Gain Control (AGC) & Automatic Frequency Control (AFC)

• AGC:

  • Uses rectified detector output to generate bias voltage controlling RF/IF amplifier gain.

  • Maintains constant output despite signal strength variations.

• AFC:

  • Uses discriminator output (proportional to frequency error) to control LO frequency (via varactor).

  • Stabilizes receiver against drift.

VI. Additional Key Topics

A. Gate Function

Definition:

\[ \text{rect}\left(\frac{t}{\tau}\right) = \begin{cases} 1, & |t| \leq \tau/2 \\ 0, & \text{otherwise} \end{cases} \]

Fourier Transform:

\[ \mathcal{F}\left\{\text{rect}\left(\frac{t}{\tau}\right)\right\} = \tau \cdot \text{sinc}(f\tau), \quad \text{sinc}(x) = \frac{\sin(\pi x)}{\pi x} \]

B. Carson's Rule Applications

• Quick bandwidth estimation for FM signals.

• Example: \(f_m = 15 \text{ kHz}\), \(\Delta f = 75 \text{ kHz}\) (FM broadcast) → \(BW = 2(75+15) = 180 \text{ kHz}\) (actual FM channel = 200 kHz).

C. FM Transmitter

Block Diagram:

Modulator (direct/indirect) → Power Amplifier → Antenna.

• Direct: Varactor in oscillator.

• Indirect: Phase modulator + frequency multipliers.

D. Applications of Modulation Schemes

Final Exam Strategy:

• Fourier Transform: Practice transforms of basic signals and property applications.

• AM Power Saving: Derive formulas for DSB-SC and SSB-SC.

• SSB Generation: Phase discrimination (phasing method) diagram is frequent.

• FM: Carson’s rule, NBFM vs WBFM spectrum, PLL demodulation.

• Noise: Figure of merit derivations (AM vs DSB-SC), FM threshold effect.

• Receivers: Superheterodyne image frequency calculation, IF choice reasons.

• VSB & AGC: Short notes common (4m).