Analog & Digital Communication (IT-404) - Unit 2 Short Notes

UNIT 2: ANALOG COMMUNICATION

1. AMPLITUDE MODULATION (AM)

1.1 Introduction to AM

• Definition: Modulation process where the amplitude of a high-frequency carrier wave is varied in proportion to the instantaneous amplitude of a low-frequency message signal.

• Necessity:

  • Enables transmission over long distances ( antenna size ∝ λ ).

  • Facilitates multiplexing (FDM).

  • Overcomes limitations of low-frequency baseband signals.

• Single-tone AM Signal:

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

where:

• $$\displaystyle A_c $$ = carrier amplitude

• $$\displaystyle f_c $$ = carrier frequency

• $$\displaystyle f_m $$ = message frequency

• $$\displaystyle m_a $$ = modulation index ($$\displaystyle 0 \leq m_a \leq 1 $$ for no distortion)

• Frequency Spectrum:

  • Carrier component at $$\displaystyle f_c $$.

  • Upper Sideband (USB) at $$\displaystyle f_c + f_m $$.

  • Lower Sideband (LSB) at $$\displaystyle f_c - f_m $$.

  • Bandwidth $$\displaystyle BW = 2f_m $$.

[!TIP] Linear vs Over Modulation

• Linear: $$\displaystyle 0 < m_a < 1 $$ → no distortion.

• Over-modulation: $$\displaystyle m_a > 1 $$ → envelope distortion, carrier phase reversal.

1.2 Power Relations in AM

• Total Power:

$$ P_T = P_c \left(1 + \frac{m_a^2}{2}\right) $$

where $$\displaystyle P_c = \frac{A_c^2}{2R} $$ (carrier power, $R$ = load resistance).

• Sideband Power:

$$ P_{SB} = P_c \cdot \frac{m_a^2}{2} $$

• Power Efficiency (Transmission Efficiency):

$$ \eta = \frac{P_{SB}}{P_T} = \frac{m_a^2}{2 + m_a^2} \times 100\% $$

• Maximum $$\displaystyle \eta = 33.33\% $$ at $$\displaystyle m_a = 1 $$.

• Low efficiency because carrier consumes most power even with no information.

• Improvement: Use suppressed-carrier techniques (DSB-SC, SSB-SC) to eliminate carrier power.

1.3 AM Generation Methods

1.4 AM Demodulation

• Envelope Detector:

  • Circuit: Diode → RC low-pass filter.

  • Operation: Diode rectifies AM wave; RC filter tracks envelope.

  • Condition for no distortion: $$\displaystyle m_a \leq 1 $$ and $RC$ time constant satisfies:

$$ \frac{1}{\omega_c} \ll RC \ll \frac{1}{\omega_m} $$

• Advantage: Simple, no carrier synchronization needed.

• Synchronous (Coherent) Detector:

  • Principle: Multiply received AM by locally generated carrier (phase-locked), then low-pass filter.

  • Advantage: Works for low $$\displaystyle m_a $$ and suppressed-carrier signals; better noise immunity.

1.5 Suppressed Carrier AM Techniques

[!TIP] Comparative Analysis

• Bandwidth: SSB-SC most efficient ($$\displaystyle f_m $$), AM/DSB-SC least ($$\displaystyle 2f_m $$).

• Power: SSB-SC > DSB-SC > AM.

• Complexity: SSB-SC generation is challenging; AM simplest.

• Applications: AM for simplicity/broadcast; SSB for bandwidth/power-limited links.

2. ANGLE MODULATION (FM AND PM)

2.1 Fundamentals of FM and PM

• FM (Frequency Modulation):

  • Instantaneous frequency $$\displaystyle f_i(t) = f_c + k_f m(t) $$.

  • Modulation Index: $$\displaystyle \beta = \frac{\Delta f}{f_m} $$, where $$\displaystyle \Delta f = k_f A_m $$ (peak frequency deviation).

• PM (Phase Modulation):

  • Instantaneous phase $$\displaystyle \phi_i(t) = \omega_c t + k_p m(t) $$.

  • Modulation Index: $$\displaystyle \beta = k_p A_m $$.

• Single-tone FM Signal:

$$ s(t) = A_c \cos\left(2\pi f_c t + \beta \sin 2\pi f_m t\right) $$

• Spectrum of FM:

  • Bessel function expansion:

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

• Sidebands at $$\displaystyle f_c \pm n f_m $$; amplitudes $$\displaystyle \propto J_n(\beta) $$.

• Bandwidth: Infinite theoretically, but practical bandwidth determined by Carson's Rule:

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

• Significance:

  • Constant envelope → efficient power amplification.

  • Superior noise immunity (threshold effect, capture effect).

  • Larger bandwidth requirement.

2.2 FM Generation Techniques

2.3 Pre-emphasis and De-emphasis

• Need: FM has noise power proportional to frequency² → high-frequency SNR degradation.

• Pre-emphasis (Transmitter):

  • Boost high frequencies before modulation.

  • Transfer function: $$\displaystyle H_{pre}(f) = 1 + j\frac{f}{f_0} $$ (high-pass differentiator).

  • $$\displaystyle f_0 $$ = 2125 Hz (standard for FM broadcast).

• De-emphasis (Receiver):

  • Complementary low-pass to restore original spectrum.

  • Transfer function: $$\displaystyle H_{de}(f) = \frac{1}{1 + j\frac{f}{f_0}} $$.

  • RC circuit: $$\displaystyle R = 75\Omega $$, $$\displaystyle C = 0.1\mu F $$ (gives $$\displaystyle f_0 \approx 2122 $$ Hz).

• Net Effect: Improves output SNR for high-frequency components by ~15 dB.

2.4 Deviation Ratio and Modulation Index

• Deviation Ratio (for multi-tone FM):

$$ D = \frac{\Delta f_{max}}{f_{m,max}} $$

• $$\displaystyle \Delta f_{max} $$ = maximum frequency deviation.

• $$\displaystyle f_{m,max} $$ = highest modulating frequency.

• Significance:

  • Determines bandwidth via Carson’s rule: $$\displaystyle BW \approx 2f_{m,max}(D + 1) $$.

  • In FM broadcasting: $$\displaystyle D = 5 $$ (max deviation 75 kHz, max $$\displaystyle f_m $$ 15 kHz).

  • Higher $D$ → better noise immunity but larger bandwidth.

2.5 FM Detectors

[!TIP] Key Point: All FM detectors convert frequency deviation to amplitude variation. Preceded by limiter to remove amplitude noise.

3. RECEIVERS FOR ANALOG COMMUNICATION

3.1 Superheterodyne Receiver

• Block Diagram:

  
  Antenna → RF Amplifier → Mixer → IF Amplifier → Detector → Audio Amplifier → Speaker
  
                ↑               ↑
  
            Local Oscillator (LO)
  
  

• Frequency Conversion:

$$ f_{IF} = |f_{RF} - f_{LO}| $$

Standard IF: 455 kHz (AM), 10.7 MHz (FM).

• Image Frequency Problem:

  • Undesired frequency $$\displaystyle f_{image} = f_{RF} + 2f_{IF} $$ (or $$\displaystyle f_{RF} - 2f_{IF} $$) also produces $$\displaystyle f_{IF} $$ after mixing.

  • Image Rejection Ratio (IRR): $$\displaystyle \text{IRR} = \frac{\text{Gain at } f_{RF}}{\text{Gain at } f_{image}} $$.

  • Reduced by RF amplifier with good selectivity before mixer.

• Advantages:

  • High gain (IF amplifier fixed-tuned).

  • Excellent selectivity (narrow IF filters).

  • Uniform gain over tuning range.

  • Good stability (LO and IF fixed).

3.2 TRF Receiver

• Block Diagram:

  
  Antenna → RF Amplifier (tuned) → Tuned RF Stages → Detector → Audio Amplifier → Speaker
  
  

• Operation: All RF stages tuned to desired frequency; detector recovers baseband.

• Limitations:

  • Instability: Multiple tuned RF stages at high frequency → oscillation prone.

  • Poor Selectivity: Q of RF circuits low at high $$\displaystyle f_c $$ → adjacent channel interference.

  • No Image Rejection: No frequency conversion.

  • Variable Sensitivity: Tuning across band requires ganged tuning; gain varies with frequency.

• Conclusion: Largely obsolete; superheterodyne superior.

3.3 Receiver Performance Parameters

[!TIP] Exam Focus: Superheterodyne’s image frequency formula $$\displaystyle f_{image} = f_{RF} \pm 2f_{IF} $$ is frequently asked.

4. SAMPLING THEOREM

4.1 Nyquist Sampling Theorem for Low-Pass Signals

• Statement: A band-limited signal with maximum frequency $$\displaystyle f_{max} $$ can be completely recovered from its samples if sampled at $$\displaystyle f_s \geq 2f_{max} $$.

• Proof Sketch (Fourier Transform):

  • Sampling in time → replication in frequency (periodic spectrum with period $$\displaystyle f_s $$).

  • If $$\displaystyle f_s < 2f_{max} $$, replicas overlap → aliasing.

  • If $$\displaystyle f_s \geq 2f_{max} $$, no overlap → original spectrum recoverable by low-pass filter.

• Aliasing:

  • Consequence: High-frequency components masquerade as low-frequency → irreversible distortion.

  • Avoidance: Use anti-aliasing filter (low-pass with $$\displaystyle f_c \geq f_{max} $$) before sampling.

4.2 Sampling of Band-Pass Signals

• Signal: Bandwidth $B$, highest frequency $$\displaystyle f_H $$, lowest $$\displaystyle f_L $$ ($$\displaystyle f_H > 2B $$).

• Minimum Sampling Frequency:

$$ f_s \geq 2B \quad \text{(not necessarily } 2f_H \text{)} $$

• Conditions: $$\displaystyle f_s $$ can be as low as $2B$ if sampling rate is synchronized with band location (undersampling).

• Benefit: Reduces sampling rate for high-frequency but narrowband signals (e.g., radio IF).

4.3 Practical Considerations

• Role in Pulse Modulation:

  • PAM (Pulse Amplitude Modulation): Samples of analog signal transmitted as pulse amplitudes.

  • PCM (Pulse Code Modulation): Samples quantized and encoded into digital bits.

• Anti-aliasing Filter: Essential real-world implementation; roll-off characteristics affect required $$\displaystyle f_s $$.

• Sampling Rate Selection: $$\displaystyle f_s = 2f_{max} $$ theoretically sufficient; in practice, higher (e.g., audio CD: $$\displaystyle f_s = 44.1 $$ kHz for $$\displaystyle f_{max} = 20 $$ kHz) to allow filter roll-off.

5. ANALOG PULSE MODULATION AND MULTIPLEXING

5.1 Pulse Amplitude Modulation (PAM)

• Generation:

  • Natural Sampling: Sample-and-hold circuit; pulses have same width as sampling instant → top follows $m(t)$.

  • Flat-top Sampling: Sample-and-hold + circuit to produce flat-top pulses (easier to handle, requires reconstruction filter).

• Waveform: Amplitude of pulses proportional to $m(t)$ at sampling instants.

• Bandwidth: Approximately $$\displaystyle f_s $$ to $$\displaystyle 2f_s $$ (depends on pulse shape).

• Advantages: Simple generation.

• Disadvantages: Noise susceptibility (amplitude variations directly affect signal).

5.2 Pulse Width Modulation (PWM)

• Generation: Compare $m(t)$ with high-frequency sawtooth carrier.

  • Pulse width $\propto$ instantaneous amplitude of $m(t)$.

  • Amplitude and frequency of pulses constant.

• Waveform: Constant amplitude, variable width.

• Applications: Control systems (motor speed), telemetry, power regulation.

• Noise Immunity: Better than PAM (noise affects edges less than amplitude).

5.3 Pulse Position Modulation (PPM)

• Generation: Convert PWM (leading edge fixed, trailing edge varies) or direct timing from $m(t)$.

• Waveform: Constant amplitude and width; position (timing) of pulse varies with $m(t)$.

• Comparison:

  • Noise Immunity: Best among PAM, PWM, PPM (noise affects amplitude/width less than position if synchronized).

  • Bandwidth: Larger than PAM (requires precise timing).

  • Complexity: Demodulation requires precise timing reference.

5.4 Time Division Multiplexing (TDM)

• Synchronous TDM:

  • Each signal gets a fixed time slot in a repeating frame.

  • Frame structure: $N$ slots for $N$ signals; each slot contains one sample from corresponding signal.

  • Synchronization: Transmitter and receiver clocks must align; framing bits used.

• Block Diagram:

  
  Multiplexer: [Signal1 → Switch → Frame] + [Signal2 → Switch] + ...
  
  Demultiplexer: [Frame → Switch → Signal1] + [Switch → Signal2] + ...
  
  

• Applications: PCM telephony (E1/T1), digital audio, sensor networks.

• vs Asynchronous TDM: Statistically assigns slots based on demand; more efficient but needs buffer and addressing.

5.5 Frequency Division Multiplexing (FDM)

• Concept: Each signal modulated to a different carrier frequency; spectra placed adjacent without overlapping.

• Process:

  1. Modulate each baseband signal (AM, FM, etc.) to different carriers.

  2. Sum all modulated signals → composite FDM signal.

  3. At receiver, bandpass filters separate carriers → demodulate each.

• Application: Analog radio/TV broadcasting, cable TV, satellite transponders.

• Relation to Modulation: Each channel uses analog modulation (typically AM for video, FM for audio in TV).

[!TIP] PAM vs PWM vs PPM: Remember: PAM = amplitude varies; PWM = width varies; PPM = position varies. Noise immunity: PPM > PWM > PAM.

6. PERFORMANCE ANALYSIS OF ANALOG MODULATION SYSTEMS

6.1 Bandwidth Efficiency

6.2 Power Efficiency

• AM: $$\displaystyle \eta = \frac{m_a^2}{2 + m_a^2} $$ → max 33.3%.

• DSB-SC: All power in sidebands → 100% efficient, but requires coherent detection.

• SSB-SC: All power in one sideband → 100% efficient + bandwidth saving.

• FM: Constant envelope → power in carrier + sidebands; average power independent of modulation; efficiency low for narrowband FM, but large $\beta$ spreads power among many sidebands.

6.3 Noise Performance and SNR

• AM (Envelope Detector):

  • Output SNR: $$\displaystyle \left(\frac{S}{N}\right)_{out} \propto m_a^2 \cdot \left(\frac{S}{N}\right)_{in} $$.

  • No threshold effect; SNR degrades linearly with input SNR.

• DSB-SC (Coherent):

$$ \left(\frac{S}{N}\right)_{out} = \frac{m_a^2}{4} \left(\frac{S}{N}\right)_{in} \quad \text{(for single-tone)} $$

• 3 dB worse than AM for same $$\displaystyle m_a $$? Actually, DSB-SC has no carrier → better power efficiency but same noise figure? Correction: For same total transmitted power, DSB-SC has higher output SNR because all power is in sidebands. For same carrier power, AM has higher SNR due to carrier gain.

• SSB-SC (Coherent):

$$ \left(\frac{S}{N}\right)_{out} = \frac{m_a^2}{2} \left(\frac{S}{N}\right)_{in} $$

• 3 dB better than DSB-SC (half bandwidth → half noise power).

• FM:

  • Threshold Effect: Below certain input SNR, noise dominates → rapid SNR degradation.

  • Above threshold: Output SNR $$\displaystyle \propto \beta^2 \left(\frac{S}{N}\right)_{in}^2 $$ → improves with modulation index.

  • Capture Effect: Strongest signal captured; suppresses weaker signals → better in multi-path fading.

6.4 Applications and Selection Criteria

7. KEY CONCEPTS AND SHORT NOTE TOPICS

7.1 Frequency Translation in Superheterodyne Receivers

• Principle: Mixing RF signal with LO produces sum and difference frequencies; IF (fixed) selected via filter.

• Equation: $$\displaystyle f_{IF} = |f_{RF} - f_{LO}| $$.

• Advantage: Allows fixed-tuned IF amplifiers with high gain and selectivity.

7.2 Modulation Index and its Effect on AM Spectrum

• $$\displaystyle m_a = \frac{A_{max} - A_{min}}{A_{max} + A_{min}} $$.

• Effect:

  • $$\displaystyle m_a = 0 $$ → only carrier.

  • $$\displaystyle 0 < m_a < 1 $$ → carrier amplitude unchanged, sideband amplitudes $$\displaystyle \propto m_a $$.

  • $$\displaystyle m_a > 1 $$ → over-modulation → envelope distortion, sideband amplitudes saturate.

7.3 Linear vs Over Modulation in AM

• Linear: $$\displaystyle m_a \leq 1 $$ → envelope is scaled version of $$\displaystyle |1 + m_a \cos(2\pi f_m t)| $$ → no distortion.

• Over: $$\displaystyle m_a > 1 $$ → envelope crosses zero → phase reversal → distortion; demodulation fails.

7.4 Carson's Rule for FM Bandwidth

• Rule: $$\displaystyle BW \approx 2(\Delta f + f_m) $$.

• Justification: Contains ~98% power; accounts for significant sidebands (up to $|n| \approx \beta + 1$).

• Example: FM broadcast: $$\displaystyle \Delta f = 75 $$ kHz, $$\displaystyle f_m = 15 $$ kHz → $BW \approx 180$ kHz (vs theoretical infinite).

7.5 Capture Effect in FM

• Phenomenon: In FM reception, when multiple signals on same channel, receiver locks onto strongest signal and suppresses others.

• Cause: Limiter action + discriminator characteristic.

• Advantage: Reduces co-channel interference; improves reception in fading.

7.6 Image Frequency and Its Rejection

• Image Frequency: $$\displaystyle f_{image} = f_{RF} + 2f_{IF} $$ (or $$\displaystyle f_{RF} - 2f_{IF} $$) that also downconverts to $$\displaystyle f_{IF} $$.

• Rejection:

  • Preselector (RF amplifier) with narrow bandwidth.

  • Image Rejection Ratio (IRR) > 40 dB typical.

  • Higher IF reduces image problem (but mixer performance degrades at high IF).

7.7 Anti-aliasing Filter in Sampling

• Purpose: Band-limit input signal to $$\displaystyle f_{max} < f_s/2 $$ before A/D converter.

• Type: Analog low-pass filter (e.g., Butterworth, Chebyshev).

• Spec: Roll-off steep enough to attenuate above $$\displaystyle f_s/2 $$ to prevent aliasing.

• Consequence of omission: Aliasing → irreversible distortion.

7.8 Synchronous vs Asynchronous TDM

7.9 Trade-offs in Receiver Design (Sensitivity vs Selectivity vs Fidelity)

• Sensitivity vs Selectivity: High gain (sensitivity) may cause overload → reduces selectivity (intermodulation).

• Selectivity vs Fidelity: Narrow IF bandwidth improves selectivity but attenuates high audio frequencies → reduces fidelity.

• Optimization: Choose IF bandwidth matching signal bandwidth; use multiple IF stages; AGC for sensitivity without overload.

[!TIP] Final Exam Strategy: For short notes, define term, give formula/equation, explain significance, and mention one application. Always box key formulas. Compare/contrast in tables where applicable.