Himanish Agarwal ❚
Notes on software, systems, and the science of making machines learn

Communication Systems

[Sat Dec 17, 2022]
— Himanish

Introduction #

Major Functional Blocks #

  • Transducer: converts nonelectric message to electric signal

Analog to Digital #

  • Signal distortion increases with distance
  • Digital signals have enhanced immunity to noise and interferences
    • A finite alphabet makes the the receiver’s decision more certain
  • Analog systems: signals and noise within same BW can’t be separated
  • The sampling theorem states that if the highest frequency in the signal spectrum is B (in hertz), the signal can be reconstructed from its discrete samples, taken uniformly at a rate above 2B samples per second
  • A quantizer partitions the signal range into L intervals. Each sample amplitude is approximated by the midpoint of the interval in which the sample value falls

Channel Parameters #

Bandwidth and Power #

  • Channel bandwidth B and the signal power \(P_s\) control the connection’s rate and quality
  • The faster a signal changes, the higher its maximum frequency is, and the larger its bandwidth is
  • Increasing \(P_s\)strengthens the signal pulse and suppresses the effect of channel noise and interference, to maintain a minimum SNR (signal-noise-ratio) over a longer distance

Capacity #

\[C = B \log_2(1+\text{SNR})\] It is impossible to transmit at a rate higher than C without incurring a large number of errors.

Amplitude Modulation (AM) #

  • Tone modulation: Modulation signal contains single frequency e.g. pure sinusoidal, so impulse arrows instead of continuous spectrum

Baseband versus Carrier #

  • Baseband. freq band of original message before modulation, measured close from zero. Much lower freq than modulated signal.
  • Carrier has high frequency. Sinusoidal carrier can be formed from AM, FM or PM. (amplitude, freq, phase modulations)

Double-Sideband (DSB) Supressed Carrier AM #

Carrier frequency should be greater than bandwidth of modulating signal

Synchronous/Coherent Demodulation #

Demodulation can be done by repeating the modulation process, and the original message can be obtained via a LPF. But it requires a signal with same frequency as carrier, which is difficult. Signal attenuates/time delay so receiver complexity increases. This is fine for point to point but not broadcast.

Conventional AM (DSB Full Carrier) #

  • The carrier is sent with the modulated signal. This can be done by a dc offset \(A_c\) before modulation.
    • \(|A_c + m(t)| \geq 0\) to avoid zero crossing to prevent phase reversal in freq domain, which distorts the envelope
  • By following the envelope, we can recover the original signal
    • \(f_c » f_m\) (max freq of message signal)

Envelope Detection #

  • Diode removes negative half
  • RC circuit slowly discharges to follow the envelope
  • \(2\pi B < \frac{1}{RC} < \omega_c\)
  • \(RC \le \frac{1}{\omega_c}\frac{\sqrt{1-\mu^2}}{\mu}\)

Modulation Index #

  • \[\mu = \frac{A_m}{A_c}\]

    • \(\mu < 1\) to prevent overmodulation
    • \(A_c = \frac{V_{max} + V_{min}}{2}\)
    • \(A_m = \frac{V_{max} - V_{min}}{2}\)

    \[\mu = \frac{V_{max} - V_{min}}{V_{max} + V_{min}}\]

    • \(\mu = \frac{V_{max} - V_{min}}{2V_C + V_{max} + V_{min}}\) for non zero offset

  • For multi-tone modulation \(\mu_T = \sqrt{\mu_1^2+\cdots+\mu_n^2}\)

Power #

  • For singletone \[ P_{SB} = P_c \cdot \frac{\mu^2}{2}\]
    • \(P_{tot} = P_c (1 + \frac{\mu^2}{2})\)
  • In general, \[P_T = \frac{\overline{m^2(t)}}{2} + \frac{A_c^2}{2}\]

Efficiency #

Useful power resides in sidebands, whereas carrier power is only for convenience in mod-demod. \[\eta = \frac{P_c}{P_{c}+P_{SB}} = \frac{\mu^2}{2+\mu^2}\]

Single Sideband (SSB) #

\[\phi_{\text{USB, LSB}}(t) = m(t)\cos\omega_ct \mp m_h(t) \sin\omega_ct\]

Angle Modulation #

  • Constant amplitude hence \(P_{av} = \frac{A^2}{2}\)

  • Bandwidth required more than AM and depends on modulation index unlike AM

  • Better noise immunity than AM and can be increases with \(\Delta f\)

  • Transmitters and receivers are more complex than AM

  • All transmitted power is useful (no carrier and sidebands)

    https://www.youtube.com/watch?v=g1RiAmB1J5k

Single Tone #

  • Carrier signal \(A_c\cos\omega_ct\)
  • Freq deviation \(\Delta f = k_fA_m\)
  • Modulation index \(\beta = \frac{\Delta f}{f_m}\) \[\phi_{FM}(t) = A_c\cos(\omega_ct + \beta \sin\omega_mt)\]

NBFM (Narrowband) #

If \[\left|k_f \int_{-\infty}^t m(\alpha)d\alpha\right| \ll 1\] then \(k_f \int_{-\infty}^t m(\alpha)d\alpha \approx k_f\sin\omega_mt\)

  • Bandwidth \(B_{FM} \approx 2f_m\) comparable to AM
  • Requires \(\beta \le 0.3\) rad

WBFM (Wideband) #

\[\beta > 0.3\] \(B_{FM}^{WB} \approx 2(\Delta f + B) = 2B(\beta+1)\) [Carson’s rule]

Phase Locked Loop (PLL) #

  • Phase detector: output proportional to phase difference between inputs
  • VCO (Voltage Controlled Oscillator): monotonic frequency-vs-voltage characteristic (unstable)
  • Loop filter: removes hi-freq components
  • Stable hi-freq output using a reference lo-freq oscillator

Superheterodyne Receiver #

https://www.youtube.com/watch?v=dk6DdG4vs4Y

Digital Communication #

\[x(t) = \sum_k a_kp_T(t-kT) | T: \text{symbol duration}\]

Spectrum of Transmitted Signal #

  • Cannot find direct expectation of pulse as that would imply spectrum is zero but we need a spectrum to transmit a signal.
  • Hence, power spectral density comes in. For that we need the autocorrelation.
    • \(R_{xx}(\tau) = E{x(t)x(t+\tau)} = \frac{P_d}{T}R_{P_TP_T}(\tau)\)
    • \(= \sum_kE{a_k^2}p(t-kT)p(t+\tau-kT)\)
    • Data Symbol Power \(P_d = E{a_k^2} = A^2\)
    • Taking Fourier transform, on both sides, \[\overline{S_{xx}(f)}^{\text{PSD of x(t)}} = \frac{P_d}{T} \overline{S_{PP}(f)}^{\text{Energy Spectral Density}}\]
  • \(S_{xx}(f) = \frac{P_d}{T}|P_T(f)|^2 = P_dT sinc^2(fT)\)

AWGN #

  • Additive: \(y = x + n\)
  • White: Noise samples at any two different times are uncorrelated.
    • \(R_{NN}(\tau) = \frac{n}{2}\delta(\tau)\)
    • \(S_{NN} = \frac{n}{2}\) i.e. power spread equally across all frequencies just like white light
  • Remains Gaussian after any filtering as filter is linear

Data Coding #

Convolutional Code #

  • If size of shift register is \(n\), rate of code is \(1/n\), i.e. input sequence is \(1/n\) as long as output
  • Either send the code \(n\) times faster so BW required is \(n\) times and energy per coded bit becomes \(1/n\)th so more chance of error in a bit. But we can use parity checks to correct these errors.
  • Or send at the same rate but use a higher order modulation e.g. for 3-bit shift register use 8-PSK instead of BPSK so you can send 3 bits at once