Communication Systems Intro

Cheatsheet Content

### Introduction to Communication Systems Communication systems are vital for transmitting information from a source to a destination. They involve various components and processes to ensure efficient and reliable data transfer. #### Cheatsheet Overview This cheatsheet covers fundamental concepts in communication systems, including: - Scoring Rubrics for the course - Recommended Reference Materials - Key Topics in Communication Systems - Detailed Introduction to Communication Systems - Block Diagrams and Components - Modes of Transmission (Simplex, Half-duplex, Full-duplex) - Analog vs. Digital Signals - Modulation and Multiplexing - The Electromagnetic Spectrum - Limitations of Communication Systems (Shannon Capacity Theorem) ### Reference Materials 1. **A.B. Carlson, P.B. Crilly, J.C. Rutledge**, *Communication Systems*, McGraw Hill, 2002. 2. **L.W. Couch II**, *Digital and Analog Communication Systems*, 8th Edition, Prentice Hall, 2013. 3. **J.G. Proakis, M.Salehi**, *Fundamentals of Communications Systems*, Prentice Hall, 2005. 4. **R.E. Ziemer, W.H. Tranter**, *Principles of Communications*, Wiley, New York, 2009. ### Topics 1. Introduction of Communication Systems 2. Signal and System 3. Fourier Transform 4. Analog Signal Transmission (AM, FM, and PM) 5. Noise 6. Analog to Digital Conversion (PCM) 7. Baseband Digital Transmission (Line coding) 8. Digital Modulation (ASK, FSK, and PSK) 9. Multiplexing Techniques (FDM, TDM, CDM, and OFDM) 10. Advanced Communication Systems ### Introduction of Communication Systems - **Overview of communication systems**: Understanding the basic structure and components. - **Communication systems block diagram**: Visual representation of signal flow. - **Modes of Transmission**: Simplex, Half-duplex, and Full-duplex. - **Analog VS Digital**: Comparison of signal types. - **Modulation & Multiplexing**: Techniques for signal preparation and simultaneous transmission. - **The electromagnetic spectrum**: Frequency allocation for various communication types. - **Limitation of Communication System**: Factors affecting system performance. #### Communication System Block Diagram A typical communication system consists of: - **Information**: The message to be sent (analog/digital). - **Transducer**: Converts information into a suitable electrical signal (e.g., microphone). - **Transmitter**: Processes the signal for transmission over a channel (e.g., modulation). - **Channel**: The medium through which the signal travels (wired or wireless). - **Noise**: Unwanted signals that interfere with the transmitted information. - **Receiver**: Reconstructs the original signal from the channel (e.g., demodulation). #### Modes of Transmission - **Simplex (SX)**: Communication in one direction only (e.g., broadcast radio). - Sender → Receiver - **Half-duplex (HDX)**: Communication in both directions, but not simultaneously (e.g., walkie-talkie). - Sender ↔ Receiver (alternating) - **Full-duplex (FDX)**: Communication in both directions simultaneously (e.g., telephone). - Sender ↔ Receiver (simultaneous) #### Analog VS Digital - **Analog Data**: Continuous data that varies smoothly over time (e.g., sound, video, sensor readings like temperature). - Amplitude changes continuously. - **Digital Data**: Discrete data with abrupt changes in signal level, often represented by square waves (e.g., text, binary code). - Amplitude is limited to specific levels (e.g., 1 for positive voltage, 0 for negative voltage). #### Comparison: Analog vs. Digital Signals | Feature | Analog | Digital | |--------------|--------------------------------------------|----------------------------------------------| | **Purpose** | Primarily for audio | For both data and audio | | **Noise** | Easily affected by noise, prone to errors | Less affected by noise, robust to errors | | **Speed** | Relatively low data transmission speed | High data transmission speed | | **Design** | Simpler circuit design | More complex circuit design | #### Modulation & Multiplexing - **Modulation**: The process of mixing an information signal with a high-frequency carrier signal. The carrier transports the information over long distances via wired or wireless media. - **Purpose**: To shift the signal's frequency to an appropriate range for transmission. - **Multiplexing**: Combining multiple information signals for simultaneous transmission over a shared communication channel. - **Types**: FDM (Frequency Division Multiplexing), TDM (Time Division Multiplexing), WDM (Wavelength Division Multiplexing), CDM (Code Division Multiplexing), OFDM (Orthogonal Frequency Division Multiplexing). #### The Electromagnetic Spectrum The electromagnetic spectrum is divided into various frequency bands, each used for different communication applications. - **Radio Waves**: ELF, VF, VLF, LF, MF, HF, VHF, UHF, SHF, EHF. - **Optical Spectrum**: Infrared, Visible Light, Ultraviolet. - **Higher Frequencies**: X-rays, Gamma rays, Cosmic rays. - **Frequency and Wavelength Relationship**: - Wavelength ($\lambda$) = Speed of light ($c$) / frequency ($f$) - Speed of light ($c$) $\approx 3 \times 10^8$ meters/second - Therefore: $\lambda = c/f$ #### Limitation of Communication System System performance is limited by: - **Power**: Transmitter power. - **Size**: Physical size of components. - **Gain**: Amplification of signals. - **Bandwidth**: The range of frequencies available for transmission. - **Shannon Capacity Theorem**: Defines the maximum rate at which information can be transmitted over a noisy channel. $$C = BW \log_2(1 + SNR) \text{ bps}$$ Where: - $C$: Channel Capacity (bits per second) - $BW$: Bandwidth of the channel (Hertz) - $SNR$: Signal-to-Noise Ratio (dimensionless) ### Signals and Systems #### Signal Definition A signal is a physical phenomenon that can be described mathematically, often as a function of one or more independent variables. In communication systems, signals carry information. - **Examples of Signals**: - Audio signals - Image signals - Electrocardiogram (ECG) signals - Blood pressure signals - Temperature changes - Stock prices #### Types of Signals Signals can be classified based on various characteristics: ##### Continuous-Time (CT) vs. Discrete-Time (DT) Signals - **Continuous-Time Signal**: A signal defined for all values of time $t$. Represented as $x(t)$. - Examples: Speech, video, radio waves. - **Discrete-Time Signal**: A signal defined only at discrete time instants $k$. Represented as $x[k]$. - Often obtained by sampling a continuous-time signal: $x[k] = x(t)|_{t=kT_s} = x(kT_s)$, where $T_s$ is the sampling period. - Examples: Stock market indices, digital audio samples. ##### Analog vs. Digital Signals - **Analog Signal**: A continuous-time signal whose amplitude can take any value within a range. - **Digital Signal**: A signal whose amplitude is quantized to a finite set of values. It can be continuous-time or discrete-time. - **Binary Signal**: A digital signal with only two amplitude levels (e.g., 0 and 1). - Data Rate ($R$) = $1/T$ bps, where $T$ is the bit duration. - Advantages: Fast processing, simple circuits, low cost. - Disadvantages: Requires many bits to represent complex information. - **Multi-level Signal**: A digital signal with more than two amplitude levels. - Advantages: More efficient data storage/transmission. - Disadvantages: More complex circuitry. ##### Periodic vs. Aperiodic Signals - **Periodic Signal**: A signal that repeats itself over a fixed interval called the period ($T$). - For a continuous-time signal: $x(t) = x(t + nT)$ for any integer $n$. - The fundamental period ($T_0$) is the smallest positive $T$ for which this holds. - Fundamental frequency ($f_0$) = $1/T_0$. - For a discrete-time signal: $x[k] = x[k + N]$ for any integer $N$. - **Aperiodic Signal**: A signal that does not repeat itself. ##### Deterministic vs. Random Signals - **Deterministic Signal**: A signal whose properties can be precisely determined and described by mathematical equations at any given time. - **Random Signal**: A signal whose properties cannot be predicted precisely and can only be described statistically (e.g., noise, information signals like speech). #### Energy and Power Signals - **Power of a Signal**: In an electrical system with resistance $R=1 \Omega$, the instantaneous power is $p(t) = |g(t)|^2$ Watts. - **Total Energy ($E$)**: The total energy of a signal $g(t)$ is given by: $$E = \lim_{T \to \infty} \int_{-T}^{T} |g(t)|^2 dt = \int_{-\infty}^{\infty} |g(t)|^2 dt \text{ Joules}$$ - An **Energy Signal** has $0 1/2 \end{cases}$$ - **Triangle Function ($\Lambda(t)$)**: A triangular pulse of width 2 and peak amplitude 1, centered at $t=0$. $$\Lambda(t) = \begin{cases} 1-|t|, & |t| - **Sinc Function ($\text{sinc}(t)$)**: Defined as $\text{sinc}(t) = \frac{\sin(\pi t)}{\pi t}$. - **Unit Step Function ($u(t)$)**: $$u(t) = \begin{cases} 1, & t > 0 \\ 1/2, & t = 0 \\ 0, & t - **Signum Function ($\text{sgn}(t)$)**: $$\text{sgn}(t) = \begin{cases} +1, & t > 0 \\ 0, & t = 0 \\ -1, & t ### Fourier Transform The Fourier Transform is a mathematical tool used to convert signals from the time domain to the frequency domain. It's crucial for analyzing signal spectra and designing filters. #### Introduction - **Purpose**: Converts time-domain signals $g(t)$ into frequency-domain signals $G(f)$. - **Signal Spectrum**: The frequency-domain representation ($G(f)$) reveals the signal's frequency content, known as the signal spectrum. This is essential for understanding filter design and other system components. - **Fourier Series**: Used for periodic signals. - **Fourier Transform**: Applicable to both periodic and aperiodic signals. #### Definition - **Forward Fourier Transform**: Converts a time-domain signal $g(t)$ to its frequency-domain representation $G(f)$. $$G(f) = \int_{-\infty}^{\infty} g(t)e^{-j2\pi ft} dt$$ - **Inverse Fourier Transform**: Converts a frequency-domain signal $G(f)$ back to its time-domain representation $g(t)$. $$g(t) = \int_{-\infty}^{\infty} G(f)e^{j2\pi ft} df$$ #### Fourier Spectrum The Fourier Transform $G(f)$ is generally a complex-valued function and can be expressed in polar form: $$G(f) = |G(f)|e^{j\theta(f)}$$ - $|G(f)|$: Continuous amplitude spectrum. - $\theta(f)$: Continuous phase spectrum. For real-valued signals $g(t)$, the following properties hold: - $G(-f) = G^*(f)$ (conjugate symmetry) - $|G(-f)| = |G(f)|$ (even symmetry for amplitude spectrum) - $\theta(-f) = -\theta(f)$ (odd symmetry for phase spectrum) #### Properties of Fourier Transform - **Linearity**: If $g_1(t) \leftrightarrow G_1(f)$ and $g_2(t) \leftrightarrow G_2(f)$, then $ag_1(t) + bg_2(t) \leftrightarrow aG_1(f) + bG_2(f)$. - **Time Shifting**: If $g(t) \leftrightarrow G(f)$, then $g(t-t_0) \leftrightarrow e^{-j2\pi ft_0} G(f)$. - **Time Scaling**: If $g(t) \leftrightarrow G(f)$, then $g(at) \leftrightarrow \frac{1}{|a|} G\left(\frac{f}{a}\right)$. - **Conjugation**: If $g(t) \leftrightarrow G(f)$, then $g^*(t) \leftrightarrow G^*(-f)$. - **Duality**: If $g(t) \leftrightarrow G(f)$, then $G(t) \leftrightarrow g(-f)$. - **Time Differentiation**: If $g(t) \leftrightarrow G(f)$, then $\frac{d^n g(t)}{dt^n} \leftrightarrow (j2\pi f)^n G(f)$. - **Frequency Differentiation**: If $g(t) \leftrightarrow G(f)$, then $(-j2\pi t)^n g(t) \leftrightarrow \frac{d^n G(f)}{df^n}$. - **Time Integration**: If $g(t) \leftrightarrow G(f)$, then $\int_{-\infty}^{t} g(\tau) d\tau \leftrightarrow \frac{1}{j2\pi f} G(f) + G(0)\delta(f)$. - **Convolution**: If $g_1(t) \leftrightarrow G_1(f)$ and $g_2(t) \leftrightarrow G_2(f)$, then $g_1(t) * g_2(t) \leftrightarrow G_1(f)G_2(f)$. - **Area under $g(t)$**: $\int_{-\infty}^{\infty} g(t) dt = G(0)$. - **Area under $G(f)$**: $\int_{-\infty}^{\infty} G(f) df = g(0)$. #### Fourier Transform Pairs Table | No. | $g(t)$ | $G(f)$ | Condition | |-----|--------|--------|-----------| | 1 | $e^{-at}u(t)$ | $\frac{1}{a+j2\pi f}$ | $a > 0$ | | 2 | $e^{at}u(-t)$ | $\frac{1}{a-j2\pi f}$ | $a > 0$ | | 3 | $e^{-a|t|}$ | $\frac{2a}{a^2+(2\pi f)^2}$ | $a > 0$ | | 4 | $te^{-at}u(t)$ | $\frac{1}{(a+j2\pi f)^2}$ | $a > 0$ | | 5 | $t^n e^{-at}u(t)$ | $\frac{n!}{(a+j2\pi f)^{n+1}}$ | $a > 0$ | | 6 | $\delta(t)$ | $1$ | | | 7 | $1$ | $\delta(f)$ | | | 8 | $e^{j2\pi f_0 t}$ | $\delta(f-f_0)$ | | | 9 | $\delta(t-t_0)$ | $e^{-j2\pi f t_0}$ | | | 10 | $\cos(2\pi f_0 t)$ | $\frac{1}{2}(\delta(f+f_0) + \delta(f-f_0))$ | | | 11 | $\sin(2\pi f_0 t)$ | $j\frac{1}{2}(\delta(f+f_0) - \delta(f-f_0))$ | | | 12 | $u(t)$ | $\frac{1}{2}\delta(f) + \frac{1}{j2\pi f}$ | | | 13 | $\text{sgn}(t)$ | $\frac{2}{j2\pi f}$ | | | 14 | $\cos(2\pi f_0 t) u(t)$ | $\frac{1}{4}(\delta(f+f_0) + \delta(f-f_0)) + \frac{j2\pi f}{(2\pi f_0)^2+(2\pi f)^2}$ | | | 15 | $\sin(2\pi f_0 t) u(t)$ | $j\frac{1}{4}(\delta(f+f_0) - \delta(f-f_0)) + \frac{2\pi f_0}{(2\pi f_0)^2+(2\pi f)^2}$ | | | 16 | $e^{-at}\cos(2\pi f_0 t) u(t)$ | $\frac{a+j2\pi f}{(a+j2\pi f)^2+(2\pi f_0)^2}$ | | | 17 | $e^{-at}\sin(2\pi f_0 t) u(t)$ | $\frac{2\pi f_0}{(a+j2\pi f)^2+(2\pi f_0)^2}$ | | | 18 | $\Pi(t/\tau)$ | $\tau \text{sinc}(f\tau)$ | | | 19 | $B \text{sinc}(Bt)$ | $\Pi(f/B)$ | | | 20 | $\Lambda(t/\tau)$ | $\tau \text{sinc}^2(f\tau)$ | | | 21 | $B \text{sinc}^2(fB)$ | $\Lambda(f/B)$ | | | 22 | $\sum_{n=-\infty}^{\infty} \delta(t-nT)$ | $f_0 \sum_{n=-\infty}^{\infty} \delta(f-nf_0)$ | | | 23 | $e^{-t^2/(2\sigma^2)}$ | $\sigma\sqrt{2\pi} e^{-2(\sigma\pi f)^2}$ | | ### Amplitude Modulation (AM) AM is a modulation technique where the amplitude of the carrier signal is varied in proportion to the message signal. #### Introduction - **Goal**: To transmit information (message signal) accurately from source to destination. - **Baseband Signal**: The message signal $m(t)$ before modulation. - **Modulation Process**: Shifts the baseband signal to a higher frequency for efficient transmission. - **Demodulation**: Recovers the original baseband signal from the modulated carrier at the receiver. #### AM Techniques 1. **Amplitude Modulation (AM)**: - AM (Amplitude Modulation) - DSB-SC (Double Sideband-Suppressed Carrier) - SSB (Single Sideband) - VSB (Vestigial Sideband) 2. **Angle Modulation**: - FM (Frequency Modulation) - PM (Phase Modulation) #### Bandwidth (BW) - **Definition**: The frequency range within which a channel can transmit information. - **Human Auditory Range**: 20-20,000 Hz. - **Standard Voice Band**: 300-3400 Hz. - **Note**: Frequency unit is Hertz (Hz) or cycles per second (cps). #### Standard AM - **Carrier Signal**: $c(t) = A_c \cos(2\pi f_c t)$ - **AM Signal**: $s_{AM}(t) = A_c[1 + m(t)]\cos(2\pi f_c t)$ - **Modulation Index ($\mu$)**: Determines the maximum percentage of variation of $|m(t)|$. - $\mu = A_m / A_c$, where $A_m$ is the amplitude of the message signal. ##### Overmodulation - **Condition for no overmodulation**: $|m(t)| 0$. - **Overmodulation**: Occurs when $|m(t)| > 1$, causing phase reversal and distortion of the envelope. ##### Spectrum and Bandwidth - **AM Signal Spectrum**: $$S_{AM}(f) = \frac{A_c}{2}[\delta(f-f_c)+\delta(f+f_c)] + \frac{A_c}{2}[M(f-f_c)+M(f+f_c)]$$ - The spectrum consists of two impulses at $\pm f_c$ (carrier components) and shifted versions of the message signal spectrum $M(f)$ (sidebands). - **AM Bandwidth ($B_{AM}$)**: $B_{AM} = 2W$, where $W$ is the bandwidth of the message signal $m(t)$. ##### Power Efficiency - **Average Power of AM Signal ($P_{AM}$)**: $$P_{AM} = \frac{A_c^2}{2}[1 + \langle m^2(t) \rangle]$$ If $\langle m(t) \rangle = 0$ (no DC component in message signal), then: $$P_{AM} = P_c (1 + \mu^2/2)$$ where $P_c = A_c^2/2$ (carrier power) and $P_m = \langle m^2(t) \rangle = \mu^2/2$ (message power). - **Efficiency ($\eta$)**: The ratio of message power to total transmitted power. $$\eta = \frac{\langle m^2(t) \rangle}{1 + \langle m^2(t) \rangle} \times 100\%$$ - Standard AM is inefficient because a significant portion of power is in the carrier, which carries no information. #### AM Signal Generation 1. **Switching Modulator**: Uses a diode (switch) to multiply the message and carrier signals. 2. **Square-Law Modulator**: Uses a non-linear device (e.g., diode, transistor) followed by a bandpass filter. #### AM Demodulation (Envelope Detector) - **Function**: Recovers the message signal $m(t)$ by detecting the envelope of the AM signal. - **Components**: Diode, resistor, and capacitor. - **Operation**: The diode rectifies the signal, and the RC circuit follows the envelope. - **RC Time Constant**: Must be chosen carefully to trace the envelope without excessive ripple or clipping. - $1/f_c \ll RC \ll 1/W$ (where $W$ is message bandwidth). #### Double Sideband-Suppressed Carrier (DSB-SC) - **Generation**: Product of message $m(t)$ and carrier $c(t)$: $s_{DSB-SC}(t) = m(t) A_c \cos(2\pi f_c t)$. - **Spectrum**: $$S_{DSB-SC}(f) = \frac{A_c}{2}[M(f-f_c) + M(f+f_c)]$$ - No carrier component (impulses) at $\pm f_c$. - **Bandwidth**: $B_{DSB-SC} = 2W$. Same as AM. - **Demodulation**: Envelope detector cannot be used due to phase reversals. Coherent detection is required. ##### DSB-SC Signal Generation 1. **Balanced Modulator**: Uses two AM modulators with opposing phases. 2. **Ring Modulator**: Uses four diodes in a ring configuration, controlled by a square-wave carrier. ##### DSB-SC Demodulation (Coherent Detector) - **Function**: Multiplies the received DSB-SC signal by a locally generated carrier that is coherent (same frequency and phase) with the original carrier, then low-pass filters. - **Costas Loop**: A feedback control system used for coherent demodulation, especially for DSB-SC and SSB signals, to maintain phase synchronization. #### Quadrature Amplitude Modulation (QAM) - **Function**: Transmits two independent message signals ($m_1(t)$ and $m_2(t)$) simultaneously over the same channel using two carriers in quadrature (90 degrees out of phase). - **QAM Signal**: $s_{QAM}(t) = A_c m_1(t)\cos(2\pi f_c t) + A_c m_2(t)\sin(2\pi f_c t)$. - **Bandwidth**: Same as DSB-SC, $2W$, but carries twice the information. #### Single Sideband (SSB) - **Function**: Transmits only one sideband (upper or lower) of the modulated signal, suppressing the carrier and the other sideband. - **Bandwidth**: $B_{SSB} = W$. This is half the bandwidth of AM and DSB-SC, making it very spectrally efficient. ##### SSB Signal Generation 1. **Filter Method**: Generate DSB-SC, then use a sharp bandpass filter to select only one sideband. 2. **Phase Shift Method**: Uses two balanced modulators and a 90-degree phase shifter for both message and carrier signals. ##### SSB Demodulation 1. **Coherent Detector**: Similar to DSB-SC, requires a local carrier synchronized in frequency and phase. 2. **Carrier Insertion Method**: Reinsert a carrier signal into the SSB signal, then use an envelope detector. This is less ideal due to potential phase mismatch. #### Vestigial Sideband (VSB) - **Function**: A compromise between DSB-SC and SSB. It transmits the carrier, one full sideband, and a "vestige" (a small part) of the other sideband. - **Advantage**: Easier filter design than SSB, as it avoids the need for ideal filters. - **Applications**: Used in television broadcasting. ### Angle Modulation Angle modulation techniques (FM and PM) vary the frequency or phase of the carrier signal in proportion to the message signal, while keeping the carrier's amplitude constant. #### Introduction - **Principle**: Varies the phase or frequency of the carrier signal. - **Constant Amplitude**: The amplitude of the carrier remains constant. - **Noise Immunity**: More robust to noise compared to AM. - **Bandwidth**: Generally requires wider bandwidths. - **Modulated Signal**: $s(t) = A_c \cos[\theta(t)]$, where $\theta(t)$ is the instantaneous phase. #### Instantaneous Frequency and Phase - **Instantaneous Frequency ($f_i(t)$)**: The rate of change of the instantaneous phase. $$f_i(t) = \frac{1}{2\pi} \frac{d\theta(t)}{dt}$$ - **Carrier Signal Phase**: For a simple carrier, $\theta(t) = 2\pi f_c t + \phi$, where $f_c$ is the carrier frequency and $\phi$ is the initial phase. #### Frequency Modulation (FM) In FM, the instantaneous frequency of the carrier is varied linearly with the message signal $m(t)$. - **Instantaneous Frequency**: $f_i(t) = f_c + k_f m(t)$, where $k_f$ is the frequency sensitivity (Hz/Volt). - **FM Signal Phase**: $\theta(t) = 2\pi f_c t + 2\pi k_f \int m(\eta) d\eta$ - **FM Signal**: $s_{FM}(t) = A_c \cos\left[2\pi f_c t + 2\pi k_f \int m(\eta) d\eta\right]$ #### Phase Modulation (PM) In PM, the instantaneous phase of the carrier is varied linearly with the message signal $m(t)$. - **PM Signal Phase**: $\theta(t) = 2\pi f_c t + k_p m(t)$, where $k_p$ is the phase sensitivity (radians/Volt). - **PM Signal**: $s_{PM}(t) = A_c \cos[2\pi f_c t + k_p m(t)]$ #### Comparison of FM and PM - If the message signal $m(t)$ is integrated before being applied to a PM modulator, the output is an FM signal. - If the message signal $m(t)$ is differentiated before being applied to an FM modulator, the output is a PM signal. #### Bandwidth of FM Signals For a sinusoidal message signal $m(t) = A_m \cos(2\pi f_m t)$: - **Frequency Deviation ($\Delta f$)**: The maximum change in carrier frequency. $$\Delta f = k_f A_m$$ - **Modulation Index ($\beta$)**: For FM, $\beta = \frac{\Delta f}{f_m}$. - **FM Signal Phase**: $\theta(t) = 2\pi f_c t + \beta \sin(2\pi f_m t)$ - **FM Signal**: $s_{FM}(t) = A_c \cos[2\pi f_c t + \beta \sin(2\pi f_m t)]$ ##### Carson's Rule An approximation for the bandwidth of an FM signal: $$B_{FM} \approx 2(\Delta f + f_m) = 2f_m(\beta+1)$$ ##### Bessel Functions For wideband FM, the spectrum consists of an infinite number of sidebands with amplitudes given by Bessel functions of the first kind, $J_n(\beta)$. #### FM Signal Generation 1. **Direct Method**: Uses a Voltage-Controlled Oscillator (VCO) whose frequency is directly varied by the message signal. - Advantages: Simple, high frequency deviation possible. - Disadvantages: Carrier frequency stability issues (often requires a Phase-Locked Loop - PLL). 2. **Indirect Method (Armstrong Method)**: Starts with a narrow-band PM signal, then converts it to FM and uses frequency multipliers and mixers to achieve the desired frequency deviation and carrier frequency.