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 Ps 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 Psstrengthens 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=Blog2(1+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 Ac before modulation.
    • |Ac+m(t)|≥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
    • fc»fm (max freq of message signal)

Envelope Detection #

  • Diode removes negative half
  • RC circuit slowly discharges to follow the envelope
  • 2πB<1RC<ωc
  • RC≤1ωc√1−μ2μ

Modulation Index #

  • μ=AmAc

    • μ<1 to prevent overmodulation
    • Ac=Vmax+Vmin2
    • Am=Vmax−Vmin2

    μ=Vmax−VminVmax+Vmin

    • μ=Vmax−Vmin2VC+Vmax+Vmin for non zero offset

  • For multi-tone modulation μT=√μ21+⋯+μ2n

Power #

  • For singletone PSB=Pc⋅μ22
    • Ptot=Pc(1+μ22)
  • In general, PT=¯m2(t)2+A2c2

Efficiency #

Useful power resides in sidebands, whereas carrier power is only for convenience in mod-demod. η=PcPc+PSB=μ22+μ2

Single Sideband (SSB) #

ϕUSB, LSB(t)=m(t)cosωct∓mh(t)sinωct

Angle Modulation #

  • Constant amplitude hence Pav=A22

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

  • Better noise immunity than AM and can be increases with Δf

  • 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 Accosωct
  • Freq deviation Δf=kfAm
  • Modulation index β=Δffm ϕFM(t)=Accos(ωct+βsinωmt)

NBFM (Narrowband) #

If |kf∫t−∞m(α)dα|≪1

then kf∫t−∞m(α)dα≈kfsinωmt

  • Bandwidth BFM≈2fm comparable to AM
  • Requires β≤0.3 rad

WBFM (Wideband) #

β>0.3

BWBFM≈2(Δf+B)=2B(β+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)=∑kakpT(t−kT)|T: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.
    • Rxx(τ)=Ex(t)x(t+τ)=PdTRPTPT(τ)
    • =∑kEa2kp(t−kT)p(t+τ−kT)
    • Data Symbol Power Pd=Ea2k=A2
    • Taking Fourier transform, on both sides, ¯Sxx(f)PSD of x(t)=PdT¯SPP(f)Energy Spectral Density
  • Sxx(f)=PdT|PT(f)|2=PdTsinc2(fT)

AWGN #

  • Additive: y=x+n
  • White: Noise samples at any two different times are uncorrelated.
    • RNN(τ)=n2δ(τ)
    • SNN=n2 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/nth 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