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Chapter 1: Introduction

Basic Communication System

• Goal: Transfer information reliably and efficiently from a Source to a Destination across a Channel.
• Fundamental Blocks: Source $\to$ Transmitter $\to$ Channel $\to$ Receiver $\to$ Destination.
• Transmitter: Encodes source information into a signal suitable for the channel (modulation).
• Channel: The physical medium (wire, fiber, air, etc.) that carries the signal. Introduces noise, distortion, attenuation.
• Receiver: Processes the received (corrupted) signal to estimate the original information (demodulation, decoding).
• Transfer Medium: Can be across space (real-time transmission) or time (storage).

The Channel: Resource and Challenge

• The channel is the resource enabling communication but presents challenges.

Channel Challenge: Limited Spectrum/Bandwidth

• Physical media cannot support infinite frequencies.
• Wireless spectrum is a scarce, regulated resource.
• Wired channels exhibit frequency-dependent attenuation (e.g., high frequencies attenuated more in twisted pairs).
• Motivation: Leads to the need for bandwidth-efficient communication techniques.

Channel Challenge: Noise

• Unavoidable random perturbations corrupting the signal.
• Thermal Noise: Due to random motion of electrons in conductors. Fundamental limit.
• Other Sources: Interference (other transmitters), man-made noise (electronics), shot noise.
• AWGN Model: Standard mathematical model. Assumes noise is:
  • Additive: $y(t)=s(t)+n(t)$.
  • White: Flat Power Spectral Density (PSD) $S_n(f)=N_0/2$ (two-sided). Noise power in bandwidth B is $P_N=N_{0}B$.
  • Gaussian: Noise amplitude follows a Gaussian distribution.
• Motivation: Need for techniques robust to noise, leading to signal design and error control coding.

Channel Challenge: Distortion

• Attenuation: Signal weakening with distance/frequency.
• Multipath (Wireless): Signal arrives via multiple paths, causing constructive/destructive interference (fading) and delay spread (causing Inter-Symbol Interference - ISI).
• Motivation: Requires equalization techniques at the receiver and robust modulation schemes.

Channel Challenge: Limited Capacity

• Shannon's Capacity Theorem: Defines the maximum theoretical rate (bits/sec) for reliable communication over a noisy channel.$$C=B{\log }_2(1+\text{SNR})$$
• Indicates a fundamental trade-off between bandwidth ($B$), power (via SNR), and data rate ($C$).
• Motivation: Benchmarks performance and drives the search for efficient coding and modulation.

Analog vs. Digital Communication

• Analog: Information represented by continuous waveforms. Simple but degrades easily with noise.
• Digital: Information represented by discrete symbols (bits). Enables sophisticated processing.
  • Why Digital? (Sec 1.1.3): Robustness (regenerative repeaters clean noise), Optimality (Source-Channel Separation allows independent optimization), Scalability (networking, packet switching), Flexibility (DSP, error correction, compression, encryption).
• Digital System Blocks: Source Coding (Compression) $\to$ Channel Coding (Error Control) $\to$ Modulation $\to$ Channel $\to$ Demodulation $\to$ Channel Decoding $\to$ Source Decoding.

Chapter 2: Signals and Systems Fundamentals

Signal Representations (Sec 2.2)

• Sinusoid: Basic periodic signal $s(t)=A\cos (2\pi f_{c}t+\theta )$.
• Complex Exponential: $s(t)=\alpha e^{j2\pi f_{c}t}$. Powerful tool, eigenfunction of LTI systems.
• Relation: $A\cos (2\pi f_{c}t+\theta )=\text{Re}\{A{e}^{j\theta }{e}^{j2\pi f_{c}t}\}$.

Inner Product, Energy, Power (Sec 2.2)

• Inner Product: $⟨s,r⟩=\int s(t)r^{\ast }(t)dt$. Measures projection/similarity.
• Energy: $E_s=\parallel s{\parallel }^2=⟨s,s⟩$.
• Power: $P_s=\overline{|s(t){|}^2}$.

Fourier Analysis (Sec 2.4, 2.5)

• Transforms signals between time and frequency domains.
• Key Pairs: $\delta (t)\leftrightarrow 1$; $e^{j2\pi f_{0}t}\leftrightarrow \delta (f-f_0)$; $\text{rect}(t/T)\leftrightarrow T\text{sinc}(fT)$.
• Convolution Theorem: $x(t)\ast h(t)\leftrightarrow X(f)H(f)$. Fundamental for LTI systems.
• Parseval's Theorem: $\int |u(t){|}^{2}dt=\int |U(f){|}^{2}df$. Relates energy in time and frequency.

Spectral Density and Bandwidth (Sec 2.6)

• ESD (Energy Signals): ${\mathcal{E}}_u(f)=|U(f){|}^2$.
• PSD (Power Signals/WSS Processes): $S_X(f)=\mathcal{F}\{R_X(\tau )\}$. Describes power distribution vs. frequency.
• Bandwidth: Defines frequency occupancy. Various definitions (e.g., 99% power containment).

Complex Baseband Representation (Sec 2.8)

• Motivation: Simplifies analysis of passband signals/systems by shifting the spectrum to baseband and using complex notation.
• Representation: Real passband $u_p(t)$ $\leftrightarrow$ Complex baseband $u(t)$.$$u_p(t)=\text{Re}\{u(t)e^{j2\pi f_{c}t}\}$$
• I/Q Components: $u(t)=u_c(t)+j{u}_s(t)$. $u_p(t)=u_c(t)\cos (2\pi f_{c}t)-u_s(t)\sin (2\pi f_{c}t)$.
• Spectrum: $U_p(f)=\frac{1}{2}[U(f-f_c)+U^{\ast }(-f-f_c)]$. The spectrum of $u(t)$ is $U(f)$.
• Power/Energy: $P_{u_p}=\frac{1}{2}{P}_u$. Conserves total energy/power (factor of 1/2 comes from Re{} operation).
• Filtering: Passband filter $H_p(f)$ corresponds to complex baseband filter $H(f)$. Output $Y(f)=\frac{1}{2}U(f)H(f)$.
• Frequency/Phase Offset: $\tilde{u}(t)=u(t)e^{j\varphi (t)}$, where $\varphi (t)=2\pi \mathrm{\Delta }ft+\gamma$. Causes rotation/mixing of I/Q components in the receiver if uncompensated. (Fig 2.28, Eq 2.77).
  
  

Chapter 3: Analog Communication Techniques

Amplitude Modulation (AM) (Sec 3.2)

DSB-SC (Double Sideband Suppressed Carrier) (Sec 3.2.1)

• Motivation: Simple modulation concept; directly maps message $m(t)$ to amplitude.
• Generation: $u_p(t)=A_{c}m(t)\cos (2\pi f_{c}t)$. Complex envelope $u(t)=A_{c}m(t)$.
• Spectrum: $U_p(f)=\frac{A_c}{2}[M(f-f_c)+M(f+f_c)]$. $BW=2B$.
• Demodulation: Requires coherent detector (multiply by local carrier $2\cos (2\pi f_{c}t+{\theta }_r)$, then LPF).
• Output: $m_{out}(t)\propto m(t)\cos {\theta }_r$.
• Challenge: Needs accurate carrier phase synchronization (${\theta }_r\approx 0$). Phase error causes signal attenuation.

Conventional AM (Sec 3.2.2)

• Motivation: Simplify receiver by enabling non-coherent envelope detection. Tradeoff: power efficiency for Rx simplicity (common in broadcasting).
• Generation: Add a large carrier component: $u_p(t)=A_c(1+a{m}_n(t))\cos (2\pi f_{c}t)$.
• $a=\frac{A|m(t{)}_{min}|}{Ac}$
• Condition for Envelope Detection: Modulation index $a\le 1$ ensures envelope $e(t)=A_c(1+a{m}_n(t))$ follows the message shape.
• Envelope Detector: Diode + RC LPF. Simple circuit.
• Power Efficiency: Inefficient due to large carrier power. ${\eta }_{AM}=\frac{a^2\overline{m_n^2}}{1+a^2\overline{m_n^2}}$. ${\eta }_{AM}\le 50\text{ Always stays in this bound}$

SSB (Single Sideband) (Sec 3.2.3)

• Motivation: Improve bandwidth efficiency over DSB by transmitting only one sideband. $BW=B$.
• Spectrum: Requires filtering DSB to remove one sideband (difficult near DC) or phase shift method.
• Hilbert Transform: Conceptual tool. $\hat{m}(t)=m(t)\ast \frac{1}{\pi t}$. Frequency domain $H(f)=-j\text{sgn}(f)$ (introduces $-{90}^{\circ }$ phase shift).
• Generation (Phase Shift Method):$$u_{USB/LSB}(t)=m(t)\cos (2\pi f_{c}t)\mp \hat{m}(t)\sin (2\pi f_{c}t)$$
  • (Scaling factors often included).
• Complex Envelope: $u_{USB/LSB}(t)=m(t)\pm j\hat{m}(t)$. (Note: Complex envelope is not just $m(t)$).
• Demodulation: Requires coherent detection. Received I component is $m(t)\cos {\theta }_r\mp \hat{m}(t)\sin {\theta }_r$.
• Challenge: Very sensitive to phase errors ${\theta }_r$, which cause not just attenuation but also distortion due to crosstalk from the Hilbert transform term.

QAM (Quadrature Amplitude Modulation) (Sec 3.2.5)

• Motivation: Transmit two independent messages ($m_c(t),m_s(t)$) in the same bandwidth as DSB ($BW=2B$) by using orthogonal carriers (cos and sin). Doubles the spectral efficiency compared to DSB/AM carrying one message.
• Generation: $u_p(t)=m_c(t)\cos (2\pi f_{c}t)-m_s(t)\sin (2\pi f_{c}t)$.
• Complex Envelope: $u(t)=m_c(t)+j{m}_s(t)$.
• Demodulation: Requires coherent receiver with two branches (I and Q).
• Challenge: Phase errors $\varphi (t)$ cause crosstalk between I and Q channels: ${\tilde{u}}_c\propto m_c\cos \varphi +m_s\sin \varphi$; ${\tilde{u}}_s\propto m_s\cos \varphi -m_c\sin \varphi$.

Angle Modulation (Sec 3.3)

• Motivation: Constant envelope $A_c$ makes it robust to amplitude nonlinearities (e.g., power amplifiers working near saturation). Can trade bandwidth for improved noise performance.
• Generation: $u_p(t)=A_c\cos (2\pi f_{c}t+\theta (t))$.
• FM: Instantaneous frequency $f_i(t)=f_c+k_{f}m(t)$. Phase $\theta (t)$ is integral of message. Smooth phase.
• PM: Phase $\theta (t)=k_{p}m(t)$. Instantaneous frequency depends on derivative of message. Can have phase discontinuities if message is discontinuous.
• Bandwidth (Carson's Rule): $B{W}_{FM}\approx 2(\mathrm{\Delta }f+B)=2B(\beta +1)$. Requires significantly more bandwidth than AM for large $\beta$.
• Demodulation (FM): Convert frequency variations to amplitude variations. E.g., Limiter-Discriminator.

Receiver Architectures (Sec 3.4)

• Superheterodyne: Mixes RF to a fixed IF for easier filtering/amplification. Requires image rejection.
• Direct Conversion (Zero-IF): Mixes directly to baseband. Simpler, but suffers from DC offset, LO leakage.

Phase Locked Loop (PLL) (Sec 3.5)

• Function: Feedback loop to align VCO output phase/frequency with input signal phase/frequency.
• Uses: Carrier synchronization (coherent demod), frequency synthesis (generating stable LO frequencies), FM demodulation.
• Linearized Analysis: Provides insight into tracking (phase steps, freq steps) based on loop filter order.

Chapter 4: Digital Modulation

Signal Constellations (Sec 4.1)

• Geometric representation of symbols in complex baseband ($b[n]=b_c[n]+j{b}_s[n]$).
• M-ary constellation carries ${\log }_{2}M$ bits per symbol.
• Examples: BPSK (2-PAM), QPSK (4-PSK/4-QAM), 8-PSK, 16-QAM. (Fig 4.4).

Bandwidth Occupancy (Sec 4.2)

• Linear Modulation: $u(t)=\sum b[n]p(t-nT)$.
• PSD (Uncorrelated Symbols): $ S_u(f) = \frac{\sigma_b^2}{T} |P(f)|^2 $. Bandwidth depends on the pulse shape $p(t)$.
• Example PSDs (Fig 4.6): Rectangular pulse has wide spectral tails (${\text{sinc}}^2$ decay); Sine pulse is better.

Design for Bandlimited Channels (Sec 4.3)

• Goal: Transmit symbols at rate $1/T$ without ISI using pulses $p(t)$ that fit within channel bandwidth.
• Nyquist Criterion for Zero ISI:
  • Time: $p(kT)=\delta [k]$. Pulse has zero crossings at multiples of symbol period $T$.
  • Frequency: $\frac{1}{T}\sum _{k}P(f-k/T)=\text{constant}$. Sum of shifted spectra is flat.
• Sinc Pulse: $p(t)=\text{sinc}(t/T)$. Achieves minimum theoretical BW $B_{min}=1/(2T)$. Impractical due to slow decay (1/t).
• Raised Cosine (RC) Pulse: Nyquist pulse with faster decay (1/t³). Introduces excess bandwidth $\alpha$.
  • $BW=(1+\alpha )B_{min}=\frac{1+\alpha }{2T}$.
  • Tradeoff: BW vs. pulse decay rate/robustness.
• Square Root Raised Cosine (SRRC): Used at Tx ($g_{TX}$) and Rx ($g_{RX}$) such that $g_{TX}(t)\ast g_{RX}(t)$ is RC. $g_{RX}(t)=g_{TX}^{\ast }(-t)$ (Matched Filter).

Bandwidth Efficiency (Sec 4.3.3)

• Measure of data rate per unit bandwidth: ${\eta }_B=\frac{R_b}{B}=\frac{({\log }_{2}M)/T}{(1+\alpha )/(2T)}=\frac{2{\log }_{2}M}{1+\alpha }$ (for complex signals/2D).
• Higher $M⟹$ higher ${\eta }_B$.

Orthogonal Modulation (Sec 4.4)

• Uses M orthogonal signals $\{s_k(t)\}$.
• FSK Example: Needs frequency separation $\mathrm{\Delta }f=1/(2T)$ (coherent) or $\mathrm{\Delta }f=1/T$ (non-coherent).
• Bandwidth: $BW\approx M\mathrm{\Delta }f$. Poor bandwidth efficiency ${\eta }_B\approx ({\log }_{2}M)/M$.

Chapter 5: Probability and Random Processes

Noise Modeling (Sec 5.8)

• AWGN: $n(t)$ with PSD $S_n(f)=N_0/2$. Autocorrelation $R_n(\tau )=\frac{N_0}{2}\delta (\tau )$.
• Noise Power in BW B: $P_N=N_{0}B$.
• Complex Baseband Noise: $n(t)=n_c(t)+j{n}_s(t)$, where $n_c,n_s$ are independent WGN processes each with PSD $N_0/2$. (Scaling adjusted for consistency). $n(t)\sim CN(0,N_0)$.

WSS Processes & Filtering (Sec 5.7, 5.9)

• LTI filtering preserves Gaussianity and (if input is WSS) WSS property.
• Output PSD: $S_Y(f)=S_X(f)|H(f){|}^2$.

Correlation and Matched Filtering (Sec 5.9)

• Matched Filter $h(t)=s^{\ast }(-t)$ maximizes SNR at output.
• Max SNR: $SN{R}_{max}=\frac{|⟨s,s⟩{|}^2}{E[|⟨n,s⟩{|}^2]}=\frac{\parallel s{\parallel }^4}{(N_0/2)\parallel s{\parallel }^2}=\frac{2\parallel s{\parallel }^2}{N_0}=\frac{2E_s}{N_0}$.

Chapter 6: Optimal Demodulation

Hypothesis Testing Framework (Sec 6.1)

• Model: Receive $Y$, choose hypothesis $H_i$ ($s_i$ sent).
• Goal: Minimize Probability of Error $P_e$.
• MAP Rule: Choose $i$ maximizing ${\pi }_{i}p(y|i)$. Optimal.
• ML Rule: Choose $i$ maximizing $p(y|i)$. Optimal for equal priors.
• Example 6.1.1: Hypothesis testing with exponential distributions.

Signal Space Concepts (Sec 6.2)

• Key Idea: Represent $M$ CT signals $\{s_i(t)\}$ as vectors $\{{\mathbf{s}}_i\}$ in an $n$-dimensional ($n\le M$) space using an orthonormal basis $\{{\psi }_k(t)\}$.
• AWGN Channel in Signal Space:$$\mathbf{Y}={\mathbf{s}}_i+\mathbf{N}$$
  • $\mathbf{Y}$: Received vector (components $Y[k]=⟨y,{\psi }_k⟩$).
  • ${\mathbf{s}}_i$: Transmitted signal vector.
  • $\mathbf{N}$: Noise vector (components $N[k]=⟨n,{\psi }_k⟩$ are i.i.d. $N(0,{\sigma }^2=N_0/2)$).
• Inner products, norms, distances are preserved.

Optimal Reception in AWGN (Sec 6.2.4)

• ML Rule (Vector Space): Choose $i$ that minimizes Euclidean distance $\parallel \mathbf{Y}-{\mathbf{s}}_i{\parallel }^2$.
• Implementation:
  • Correlator Bank: Compute $⟨y,s_i⟩$ for all $i$. Decision uses ${\text{argmax}}_i\{⟨y,s_i⟩-\parallel s_i{\parallel }^2/2\}$.
  • Matched Filter Bank: Filter $y(t)$ with $h_i(t)=s_i^{\ast }(-t)$ and sample. Output is equivalent to correlator.

Performance Analysis (Sec 6.3)

• Q-Function: $Q(x)=P[N(0,1)>x]$.
• Pairwise Error Probability (PEP): $P(i\to j)=Q\left(\frac{\parallel {\mathbf{s}}_j-{\mathbf{s}}_i\parallel }{\sqrt{2N_0}}\right)=Q\left(\frac{d_{ij}}{\sqrt{2N_0}}\right)$.
• Binary Signaling: $P_e=Q\left(\sqrt{\frac{2{\eta }_p{E}_b}{N_0}}\right)$, with ${\eta }_p=d^2/(4E_b)$.
  • Antipodal (BPSK) ${\eta }_p=1$ is 3dB better than Orthogonal/OOK (${\eta }_p=1/2$). (Fig 6.20).
• M-ary Signaling Symbol Error Rate (SER):
  • Union Bound: $P_{e|i}\le \sum _{j\ne i}Q\left(\frac{d_{ij}}{\sqrt{2N_0}}\right)$.
  • Nearest Neighbor Approx: $P_e\approx {\overline{N}}_{min}Q\left(\frac{d_{min}}{\sqrt{2N_0}}\right)$. Dominates at high SNR.
• Bit Error Rate (BER): Requires considering the bit mapping (Gray coding preferred). For Gray codes, $BER\approx SER/{\log }_{2}M$.
• Example QPSK vs 16QAM (Fig 6.25): Illustrates power vs. bandwidth tradeoff. 16QAM is more bandwidth efficient (${\eta }_B=4$ vs 2) but less power efficient (needs ~4dB higher $E_b/N_0$ for same SER).

Link Budget Analysis (Sec 6.5)

• Connects required $E_b/N_0$ (from performance analysis) to physical parameters.
• Required Received Power: $P_{RX,dBm}(\text{min})=(E_b/N_0{)}_{req,dB}+10{\log }_{10}{R}_b-174+F_{dB}$.
• Friis Free Space Path Loss: $P_{RX}=P_{TX}{G}_{TX}{G}_{RX}(\frac{\lambda }{4\pi R}{)}^2$. Relates received power to transmit power, antenna gains ($G$), range ($R$), and wavelength ($\lambda$).
• Link Budget: $P_{TX,dBm}=P_{RX,dBm}(\text{min})-G_{TX,dBi}-G_{RX,dBi}+L_{pathloss,dB}+L_{margin,dB}$.