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Communication systems and signal processing

Follow sampling, linear filters, noise, information rate, modulation, and link margins through twenty worked calculations.

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Matter pathway: atom → solid → liquid → gas → plasma. Quantum mechanics and quantum field theory provide foundations across the pathway; they are not additional phases. This is a connected modeling route, not a universal heating curve. Actual phases depend on pressure, composition, and kinetics.

1. Sampling and spectra

Definitions & inputs. fs sample frequency, B highest frequency of a real baseband signal, N sample count.

  1. Sampling replicates spectra; separation prevents overlap for generic baseband signals.

    x[n]=x(n/fs),fs>2Bx[n]=x(n/f_s),\quad f_s>2B

  2. A record of duration T sets DFT-bin spacing.

    Δf=fs/N,T=N/fs\Delta f=f_s/N,\quad T=N/f_s

  3. Choose integer m to fold a real sinusoid into the Nyquist interval.

    fa=∣f−mfs∣≤fs/2f_a=|f-mf_s|\le f_s/2

Interpretation. DFT bin spacing is not automatically the ability to resolve nearby tones; windowing matters.

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2. Linear filtering

Definitions & inputs. x[n] input, h[n] impulse response, y[n] output, H frequency response.

  1. Sum the shifted impulse responses weighted by input samples.

    y[n]=∑kh[k]x[n−k]y[n]=\sum_k h[k]x[n-k]

  2. Transform convolution into multiplication in frequency.

    H(ejω)=∑nh[n]e−jωnH(e^{j\omega})=\sum_n h[n]e^{-j\omega n}

  3. A two-sample average has cosine magnitude and a half-sample linear phase delay.

    Havg(ejω)=(1+e−jω)/2H_{avg}(e^{j\omega})=(1+e^{-j\omega})/2

Interpretation. Boundary handling and filter transient state must be defined for finite records.

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3. Noise and capacity

Definitions & inputs. k Boltzmann constant, T noise temperature, B equivalent noise bandwidth, S signal power, N noise power.

  1. Integrate the flat available noise power density across bandwidth.

    N=kTB,SNR=S/NN=kTB,\quad SNR=S/N

  2. Shannon’s AWGN limit bounds reliable rate with ideal coding and arbitrarily long blocks.

    C=Blog⁡2(1+SNR)C=B\log_2(1+SNR)

  3. Divide signal power by bit rate and noise power by bandwidth consistently.

    Eb/N0=SNR B/RbE_b/N_0=SNR\,B/R_b

Interpretation. Real coding, latency, interference and nonwhite noise change achievable performance.

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4. Digital modulation

Definitions & inputs. M constellation size, Rs symbols/s, Rb bits/s, α roll-off; γb=Eb/N0.

  1. Count bits per symbol and spectral support around the carrier.

    Rb=Rslog⁡2M,BRF=(1+α)RsR_b=R_s\log_2M,\quad B_{RF}=(1+\alpha)R_s

  2. Project AWGN onto the coherent decision axis and integrate the error tail.

    Pb=12erfc⁡(γb)P_b=\tfrac12\operatorname{erfc}(\sqrt{\gamma_b})

  3. Code rate rc reduces payload before additional framing overhead.

    Rpayload=rcRbR_{payload}=r_cR_b

Interpretation. Synchronization, fading, nonlinearities and detection method must match the BER model.

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Graphical worked example

Shannon AWGN capacity with bandwidth 1 MHz; an upper bound rather than practical throughput. X axis: Linear signal-to-noise power ratio (dimensionless). Y axis: AWGN capacity (Mbit/s).
Shannon AWGN capacity with bandwidth 1 MHz; an upper bound rather than practical throughput. Related worked calculation · Download SVG · Plot data

Twenty worked examples

Open a problem to see its defined inputs, assumptions, equation, numerical substitution, result, and interpretation. Values are illustrative analytical exercises.

Example 01. Nyquist threshold

Definitions & inputs. Real baseband content up to 20 kHz.

  1. Choose the governing model and isolate the requested quantity.

    fs,min=2Bf_{s,min}=2B

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    fs,min=2(20000)f_{s,min}=2(20000)

  3. Evaluate the expression; the result uses the units shown.

    Result=40000 samples s−1\mathrm{Result}=40000\ {\rm samples\,s}^{-1}

Interpretation. Practical sampling exceeds this threshold to allow filter transition bands.

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Example 02. Sample interval

Definitions & inputs. fs=48000 samples/s.

  1. Choose the governing model and isolate the requested quantity.

    Ts=1/fsT_s=1/f_s

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    Ts=1/48000T_s=1/48000

  3. Evaluate the expression; the result uses the units shown.

    Result=20.83333 μs\mathrm{Result}=20.83333\ {\rm \mu s}

Interpretation. This is the ideal uniform sampling period.

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Example 03. Aliased sinusoid

Definitions & inputs. Input 7 kHz, real sampling at 10 kHz.

  1. Choose the governing model and isolate the requested quantity.

    fa=∣f−fs∣f_a=|f-f_s|

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    fa=∣7−10∣f_a=|7-10|

  3. Evaluate the expression; the result uses the units shown.

    Result=3 kHz\mathrm{Result}=3\ {\rm kHz}

Interpretation. A real sampled sinusoid appears at 3 kHz with an appropriate phase.

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Example 04. DFT bin spacing

Definitions & inputs. fs=48 kHz, N=1024.

  1. Choose the governing model and isolate the requested quantity.

    Δf=fs/N\Delta f=f_s/N

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    Δf=48000/1024\Delta f=48000/1024

  3. Evaluate the expression; the result uses the units shown.

    Result=46.875 Hz\mathrm{Result}=46.875\ {\rm Hz}

Interpretation. Window leakage influences practical frequency estimation.

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Example 05. Record duration

Definitions & inputs. 1024 samples at 48 kHz.

  1. Choose the governing model and isolate the requested quantity.

    T=N/fsT=N/f_s

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    T=1024/48000T=1024/48000

  3. Evaluate the expression; the result uses the units shown.

    Result=0.02133333 s\mathrm{Result}=0.02133333\ {\rm s}

Interpretation. Zero padding does not increase this observation duration.

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Example 06. Two-tap convolution

Definitions & inputs. h=[.5,.5], x[0]=2,x[1]=4, zero before index 0; find y[1].

  1. Choose the governing model and isolate the requested quantity.

    y[1]=h[0]x[1]+h[1]x[0]y[1]=h[0]x[1]+h[1]x[0]

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    y[1]=0.5(4)+0.5(2)y[1]=0.5(4)+0.5(2)

  3. Evaluate the expression; the result uses the units shown.

    Result=3 \mathrm{Result}=3\ {}

Interpretation. This is an average of the current and preceding sample.

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Example 07. Moving-average output noise

Definitions & inputs. Independent zero-mean input noise variance 4, average 16 samples.

  1. Choose the governing model and isolate the requested quantity.

    σy2=σx2/N\sigma_y^2=\sigma_x^2/N

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    σy2=4/16\sigma_y^2=4/16

  3. Evaluate the expression; the result uses the units shown.

    Result=0.25 \mathrm{Result}=0.25\ {}

Interpretation. Correlated samples do not generally give the same reduction.

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Example 08. Decibel power ratio

Definitions & inputs. Output/input power ratio 100.

  1. Choose the governing model and isolate the requested quantity.

    G=10log⁡10(Po/Pi)G=10\log_{10}(P_o/P_i)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    G=10log⁡10100G=10\log_{10}100

  3. Evaluate the expression; the result uses the units shown.

    Result=20 dB\mathrm{Result}=20\ {\rm dB}

Interpretation. Voltage uses 20 log only when impedances are equal.

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Example 09. Power from dBm

Definitions & inputs. Received power −30 dBm.

  1. Choose the governing model and isolate the requested quantity.

    PmW=10PdBm/10P_{mW}=10^{P_{dBm}/10}

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    PmW=10−3P_{mW}=10^{-3}

  3. Evaluate the expression; the result uses the units shown.

    Result=0.001 mW\mathrm{Result}=0.001\ {\rm mW}

Interpretation. This is one microwatt.

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Example 10. Thermal noise

Definitions & inputs. T=290 K, B=1 MHz, k=1.380649×10⁻²³ J/K.

  1. Choose the governing model and isolate the requested quantity.

    N=kTBN=kTB

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    N=(1.380649×10−23)(290)(106)N=(1.380649\times10^{-23})(290)(10^6)

  3. Evaluate the expression; the result uses the units shown.

    Result=4.003882×10−15 W\mathrm{Result}=4.003882\times10^{-15}\ {\rm W}

Interpretation. Receiver added noise is excluded.

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Example 11. Noise figure cascade

Definitions & inputs. Stage1 F1=2,G1=10; stage2 F2=4, all linear power ratios.

  1. Choose the governing model and isolate the requested quantity.

    F=F1+(F2−1)/G1F=F_1+(F_2-1)/G_1

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    F=2+(4−1)/10F=2+(4-1)/10

  3. Evaluate the expression; the result uses the units shown.

    Result=2.3 \mathrm{Result}=2.3\ {}

Interpretation. Gain ahead of a noisy stage suppresses its input-referred contribution.

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Example 12. AWGN capacity

Definitions & inputs. B=1 MHz, SNR=15 linear.

  1. Choose the governing model and isolate the requested quantity.

    C=Blog⁡2(1+SNR)C=B\log_2(1+SNR)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    C=106log⁡216C=10^6\log_2 16

  3. Evaluate the expression; the result uses the units shown.

    Result=4000000 bit s−1\mathrm{Result}=4000000\ {\rm bit\,s}^{-1}

Interpretation. This is a theoretical capacity, not a modulation setting.

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Example 13. Required capacity SNR

Definitions & inputs. Target R/B=2 bit/s/Hz.

  1. Choose the governing model and isolate the requested quantity.

    SNRmin=2R/B−1SNR_{min}=2^{R/B}-1

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    SNRmin=22−1SNR_{min}=2^2-1

  3. Evaluate the expression; the result uses the units shown.

    Result=3 \mathrm{Result}=3\ {}

Interpretation. Approaching the limit requires suitable coding and large block lengths.

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Example 14. 16-QAM symbol rate

Definitions & inputs. Uncoded bit rate 8 Mbit/s.

  1. Choose the governing model and isolate the requested quantity.

    Rs=Rb/log⁡2MR_s=R_b/\log_2 M

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    Rs=8×106/4R_s=8\times10^6/4

  3. Evaluate the expression; the result uses the units shown.

    Result=2000000 symbols s−1\mathrm{Result}=2000000\ {\rm symbols\,s}^{-1}

Interpretation. Coding and framing increase the transmitted rate for a fixed payload.

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Example 15. Raised-cosine bandwidth

Definitions & inputs. Rs=1 Msymbol/s, α=.25.

  1. Choose the governing model and isolate the requested quantity.

    BRF=(1+α)RsB_{RF}=(1+\alpha)R_s

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    BRF=1.25(106)B_{RF}=1.25(10^6)

  3. Evaluate the expression; the result uses the units shown.

    Result=1250000 Hz\mathrm{Result}=1250000\ {\rm Hz}

Interpretation. This is total passband null-to-null width for the ideal pulse.

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Example 16. Bit-energy ratio

Definitions & inputs. SNR=10 linear, B=1 MHz, Rb=2 Mbit/s.

  1. Choose the governing model and isolate the requested quantity.

    Eb/N0=SNR B/RbE_b/N_0=SNR\,B/R_b

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    Eb/N0=10(1/2)E_b/N_0=10(1/2)

  3. Evaluate the expression; the result uses the units shown.

    Result=5 \mathrm{Result}=5\ {}

Interpretation. Define whether Rb is coded or information bit rate before interpreting Eb.

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Example 17. BPSK bit error probability

Definitions & inputs. Coherent AWGN channel, Eb/N0=2 linear.

  1. Choose the governing model and isolate the requested quantity.

    Pb=12erfc⁡(Eb/N0)P_b=\tfrac12\operatorname{erfc}(\sqrt{E_b/N_0})

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    Pb=12[1−erf⁡(2)]P_b=\tfrac12[1-\operatorname{erf}(\sqrt2)]

  3. Evaluate the expression; the result uses the units shown.

    Result=0.02275013 \mathrm{Result}=0.02275013\ {}

Interpretation. The result assumes perfect carrier/timing recovery and no coding.

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Example 18. Code payload rate

Definitions & inputs. Transmitted coded rate 3 Mbit/s, code rate 2/3.

  1. Choose the governing model and isolate the requested quantity.

    Rp=rcRcR_p=r_cR_c

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    Rp=(2/3)(3×106)R_p=(2/3)(3\times10^6)

  3. Evaluate the expression; the result uses the units shown.

    Result=2000000 bit s−1\mathrm{Result}=2000000\ {\rm bit\,s}^{-1}

Interpretation. Headers and pilots further reduce application payload rate.

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Example 19. Ideal ADC sine SQNR

Definitions & inputs. Full-scale sine, 12-bit uniform ideal quantizer.

  1. Choose the governing model and isolate the requested quantity.

    SQNR≃6.02N+1.76SQNR\simeq6.02N+1.76

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    SQNR≃6.02(12)+1.76SQNR\simeq6.02(12)+1.76

  3. Evaluate the expression; the result uses the units shown.

    Result=74 dB\mathrm{Result}=74\ {\rm dB}

Interpretation. Distortion, analog noise and non-full-scale amplitude reduce the measured value.

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Example 20. Link margin

Definitions & inputs. Received power −80 dBm, required sensitivity −90 dBm.

  1. Choose the governing model and isolate the requested quantity.

    M=Pr−PminM=P_r-P_{min}

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    M=−80−(−90)M=-80-(-90)

  3. Evaluate the expression; the result uses the units shown.

    Result=10 dB\mathrm{Result}=10\ {\rm dB}

Interpretation. Sensitivity must correspond to the same bandwidth, BER and modulation requirements.

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Symbols and units

Each derivation and problem defines its own symbols and inputs. Symbols may be reused with different meanings in other subjects. Keep units consistent, retain sufficient precision during calculation, and apply the stated validity limits.