The whole Communications formula sheet, to print

Holds for any random variable. $E[X^2]$ is the mean-square value (average power for a signal).

Fully described by mean $\mu$ and variance ${\sigma }^2$.

Convention used in these notes: $Q(x)=\frac{1}{2}\mathrm{erfc}(x/\sqrt{2})$. Also $Q(-x)=1-Q(x)$ and $Q(0)=0.5$.

For an ergodic process, $⟨x(t)⟩={\mu }_X$ and $⟨x^2(t)⟩=R_X(0)$.

Depends only on the lag $\tau$ when $X(t)$ is WSS. The process is real-valued here.

$A$ and $f_0$ are constants. A WSS process with zero mean and power $A^2/2$. If $\Theta$ is uniform over only $(0,\pi )$, the mean depends on $t$ and the process is not WSS.

Applies to a WSS process. $S_X(f)$ is two-sided, so it covers negative and positive frequencies.

Integrate over both sides of the spectrum. A nonzero mean contributes an impulse ${\mu }_X^2\delta (f)$ at $f=0$.

$H(0)$ is the DC gain.

Only the magnitude of $H(f)$ matters; phase has no effect on the output PSD.

The Fourier transform of this is $|H(f){|}^2{S}_X(f)$ for real $h(t)$.

Total output mean-square value, including any DC part.

Defined as $R_{YX}(\tau )=E[Y(t+\tau )X(t)]$. For the other order, $R_{XY}(\tau )=h(-\tau )\ast R_X(\tau )$ and $S_{XY}(f)=H^{\ast }(f)S_X(f)$.

Holds when $X(t)$ and $Y(t)$ are jointly WSS and uncorrelated, with at least one of them zero-mean (for example, a signal plus independent zero-mean noise). Otherwise add the cross terms $R_{XY}(\tau )+R_{YX}(\tau )$.

$X(t)$ is WSS and $\Theta \sim U(0,2\pi )$ is independent of $X(t)$. The output power is $P_Y=R_X(0)/2$.

Follows from $S_Y(f)=|H(f){|}^2{S}_X(f)$.

Samples at different instants are uncorrelated (independent if the noise is also Gaussian). Total power is infinite, so ideal white noise is a model that is used only after some filtering.

$H_0$ is the passband gain and $B$ is the one-sided bandwidth in Hz (for a low-pass filter, $|f|\le B$; for a band-pass filter, total width $B$ centred on $\pm f_c$).

Unity gain, passband $|f|\le B$, $\mathrm{sinc}(x)=\frac{\sin \pi x}{\pi x}$. $R_Y(\tau )$ is zero at $\tau =k/(2B)$, $k\ne 0$, so samples taken $1/(2B)$ apart are uncorrelated (independent if the noise is Gaussian).

Width of an ideal filter of the same peak gain that passes the same white-noise power: $P_{\text{out}}=N_0|H{|}_{\max }^2{B}_N$. The peak is at $f=0$ for a low-pass filter and at the centre frequency for a band-pass filter; $f$ in Hz.

Here $B_N=\frac{\pi }{2}{f}_{3\text{dB}}=\frac{1}{4RC}$ with $f_{3\text{dB}}=\frac{1}{2\pi RC}$. The output is coloured noise.

$k=1.38\times 10^{-23}$ J/K, $T$ in kelvin, $B$ in Hz. $\overline{v_n^2}$ is the open-circuit mean-square voltage of a resistor $R$, and $P_{\text{avail}}$ is the power it delivers to a matched load. Thermal noise is Gaussian and is white for frequencies well below $kT/h$ (about $6\times 10^{12}$ Hz at 290 K).

$n(t)$ is zero-mean, WSS and band-pass. $n_c(t)$ and $n_s(t)$ are low-pass, zero-mean and jointly WSS; if $n(t)$ is Gaussian, they are jointly Gaussian.

For white noise $N_0/2$ through an ideal band-pass filter of width $B$ (gain 1): each of the three powers equals $N_{0}B$.

For zero-mean Gaussian narrowband noise with ${\sigma }^2=E[n^2]$, written as $n(t)=r(t)\cos (2\pi f_{c}t+\psi (t))$: the envelope is Rayleigh, and the phase is uniform and independent of the envelope. $P(r>a)=\exp \left(-a^2/2{\sigma }^2\right)$ and $E[r^2]=2{\sigma }^2$. If a sinusoid of amplitude $A$ is added, the envelope becomes Rician: close to Rayleigh for $A\ll \sigma$, and close to Gaussian about $A$ for $A\gg \sigma$.

Carrier plus two sidebands, each of amplitude $\mu A_c/2$. Bandwidth $B=2f_m$ (for a general message, $B=2W$ where $W$ is the highest message frequency).

These power relations hold for any $\mu$. Without overmodulation ($\mu \le 1$) the efficiency is at most $1/3$, reached at $\mu =1$. Each sideband carries ${\mu }^2{P}_c/4$. In terms of antenna (load) current, $I_t=I_c\sqrt{1+{\mu }^2/2}$.

For a message made of tones of different frequencies, each with its own index ${\mu }_i$. Use ${\mu }_t$ in the power and efficiency formulas. The condition ${\mu }_1+{\mu }_2+\cdots \le 1$ is enough to keep the envelope from going negative.

$A_{\max }$ and $A_{\min }$ are the maximum and minimum of the envelope, for a single tone with $\mu \le 1$. For $\mu >1$ (overmodulation) the envelope is distorted and an envelope detector fails.

The first condition (with $f_m$ replaced by $W$ for a general message) lets the capacitor smooth out the carrier ripple yet follow the envelope. The second is the condition, for a single tone of index $\mu <1$, that avoids diagonal clipping (the capacitor discharging slower than the envelope falls).

Power in a $1\Omega$ load, for a zero-mean message. The spectrum has no carrier component. The envelope is $A_c|m(t)|$, not $m(t)$, so an envelope detector cannot be used; DSB-SC needs coherent detection.

DSB-SC input, local carrier of unit amplitude $\cos ({\omega }_{c}t+\varphi )$ or $\cos ({\omega }_c+\Delta \omega )t$, followed by a low-pass filter. The output vanishes at $\varphi =90^{\circ }$ (quadrature null effect), and a frequency error gives a beating (fading) output. For SSB, a phase error causes phase distortion of the output, and a frequency error shifts every output tone by $\Delta f$. A Costas loop or a squaring loop recovers a synchronous carrier for DSB-SC.

Upper sign gives USB, lower sign gives LSB. $\hat{m}(t)$ is the Hilbert transform of $m(t)$ and has the same power as $m(t)$. Power in a $1\Omega$ load. Coherent detection with a unit-amplitude local carrier gives $\frac{A_c}{4}m(t)$.

$k_p$ in rad/V, $k_f$ in Hz/V. Instantaneous frequency: FM $f_i=f_c+k_{f}m(t)$; PM $f_i=f_c+\frac{k_p}{2\pi }\frac{dm}{dt}$. FM by $m(t)$ is PM by $\int mdt$, and PM by $m(t)$ is FM by $dm/dt$.

In FM, $\Delta f$ depends only on the amplitude of the message and $\beta$ falls as $f_m$ rises. In PM, $\beta$ depends only on the amplitude and $\Delta f$ grows with $f_m$.

Spectral lines at $f_c\pm n{f}_m$ with amplitudes $A_c|J_n(\beta )|$. The carrier amplitude $A_c{J}_0(\beta )$ vanishes at $\beta \approx 2.405,5.520,8.654,\dots$

In a $1\Omega$ load. Total power is independent of $\beta$; modulation only redistributes power from the carrier to the sidebands. Power in the carrier plus $k$ pairs of sidebands is $\frac{A_c^2}{2}[J_0^2+2{\sum }_{n=1}^k{J}_n^2]$.

An approximation; for a single tone it contains roughly 98 percent of the total power. For $\beta \ll 1$, $B\approx 2f_m$ (narrowband); for $\beta \gg 1$, $B\approx 2\Delta f$ (wideband). For a general message use $B=2(\Delta f+W)=2W(D+1)$, with peak deviation $\Delta f=k_f\max |m|$ and deviation ratio $D=\Delta f/W$.

Coherent detection for DSB-SC and SSB; envelope detection for AM above threshold, with $\mu \le 1$, so its figure of merit is at most $1/3$; FM above threshold without pre-emphasis. For a general zero-mean message of power $P$: AM gives $k_a^{2}P/(1+k_a^{2}P)$ and FM gives $3k_f^{2}P/W^2$. FM beats fully modulated ($\mu =1$) AM when $\frac{3}{2}{\beta }^2>\frac{1}{3}$, that is $\beta >\sqrt{2}/3\approx 0.47$. Above threshold, FM trades bandwidth for noise performance.

The noise at an FM discriminator's output has a PSD proportional to $f^2$, so high-frequency message components suffer most. The transmitter boosts the high frequencies (pre-emphasis) and the receiver cuts them by the same amount (de-emphasis). The overall response to the message is flat, while the high-frequency noise is reduced. A $75\mu \text{s}$ time constant corresponds to $f_1\approx 2.12$ kHz.

$f_s$ is the wanted station frequency. The image station lies at $f_{LO}+f_{IF}$ and also mixes down to $f_{IF}$, causing interference. With a low-side LO ($f_{LO}=f_s-f_{IF}$), $f_{im}=f_s-2f_{IF}$.

For a single tuned circuit of loaded quality factor $Q$ at the RF stage. $\alpha$ is a voltage ratio; in dB it is $20{\log }_{10}\alpha$. With identical tuned stages before the mixer, their ratios multiply. A larger IF moves the image farther from $f_s$ and improves image rejection; a smaller IF makes adjacent-channel selectivity easier to achieve.

$B_{guard}$ is the guard band between adjacent channels. For $N$ identical SSB channels of bandwidth $W$, $B_{total}=NW+(N-1)B_{guard}$. For DSB-SC or AM channels each $B_k=2W$.

$N$ channels, each band-limited to $W$ and sampled at $f_{\text{samp}}$ (the frame rate), at or above the Nyquist rate $2W$. The minimum (Nyquist) transmission bandwidth is half the pulse rate: $N{f}_{\text{samp}}/2$ for PAM-TDM and $R_b/2$ for binary PCM-TDM. Add synchronisation bits per frame if given.

Always non-negative; a certain event ($p_i=1$) carries 0 bits.

Average information per symbol. Take $0\log 0=0$.

Minimum 0 when one symbol has probability 1; maximum ${\log }_{2}M$ when all $M$ symbols are equiprobable.

Entropy of a two-symbol source. $H_b(0.5)=1$, symmetric about $p=0.5$, $H_b(0)=H_b(1)=0$.

For the $n$th extension of a DMS (blocks of $n$ independent symbols, $M^n$ block symbols).

Chain rule. If $X$ and $Y$ are independent, $H(X,Y)=H(X)+H(Y)$. The conditional entropy is an average: $H(Y|X)={\sum }_{x}p(x)H(Y|X=x)$.

Symmetric in $X$ and $Y$; $I(X;Y)\ge 0$, with equality exactly when $X$ and $Y$ are independent.

Here $p(x,y)=p(x)P(y|x)$ and $p(y)={\sum }_{x}p(x)P(y|x)$. Terms with $p(x,y)=0$ contribute 0.

Binary code alphabet.

A binary prefix code with lengths $l_i$ exists if and only if this holds. Every uniquely decodable binary code also satisfies it, so nothing is gained in length by giving up the prefix property. It tests lengths only; a set of codewords can obey it and still not be prefix-free.

The lower bound holds for every uniquely decodable binary code; an optimal binary prefix code (e.g. Huffman) also meets the upper bound. Using the $n$th extension, $H(X)\le {\overline{L}}_n/n<H(X)+1/n$.

Binary code. For a code alphabet of $r$ symbols, $\eta =\frac{H(X)}{\overline{L}{\log }_{2}r}$. A binary code with $\overline{L}=H$ has $\eta =1$; this needs every $p_i$ to be a power of $\frac{1}{2}$.

For a channel used $r$ times per second, the capacity is $rC$ bits/s.

Reached with equiprobable inputs. $C=1$ at $p=0$ or $p=1$, and $C=0$ at $p=\frac{1}{2}$.

From $I=H(Y)-H(Y|X)$: $H(Y|X)=H_b(p)$ for every input, and $P(Y=1)=\alpha (1-p)+(1-\alpha )p$. The maximum, at $\alpha =\frac{1}{2}$, is $1-H_b(p)$.

Each bit is either received correctly or marked as erased; reached with equiprobable inputs.

From $I=H(X)-H(X|Y)$: the input is known unless the bit is erased, so $H(X|Y)=\epsilon H_b(\alpha )$. The maximum, at $\alpha =\frac{1}{2}$, is $1-\epsilon$.

Unlike discrete entropy, it can be negative. Uniform on an interval of length $a$: $h={\log }_{2}a$. Gaussian with variance ${\sigma }^2$: $h=\frac{1}{2}{\log }_2(2\pi e{\sigma }^2)$, the largest value for any density with that variance.

Band-limited AWGN channel of bandwidth $B$ Hz. $S$ is the average received signal power and $S/N$ is the power ratio (not dB); $N=N_{0}B$ with two-sided noise PSD $N_0/2$.

Convert to a ratio before using in the law: 20 dB means 100, 30 dB means 1000.

At fixed signal power $S$, capacity grows with $B$ but saturates, because the noise power $N_{0}B$ grows too.

$E_b$ is the energy per bit. With $R=C$ and $S=E_{b}C$: $\frac{E_b}{N_0}=\frac{2^{C/B}-1}{C/B}$, which tends to $\ln 2$ as $C/B\to 0$. Below this, no reliable communication is possible at any positive rate.

For a target rate $C$, a smaller bandwidth needs an exponentially larger SNR; a larger bandwidth allows a lower SNR.

Uniform quantiser with peak-to-peak range $2m_p$ and $L=2^n$ levels.

Error uniform on $(-\Delta /2,\Delta /2)$; valid when the signal is large compared with $\Delta$ and does not overload the quantiser.

Here $S=\overline{m^2}$ is the average signal power and $m_p$ the quantiser's peak value.

Sinusoid of amplitude $m_p$ so $S=m_p^2/2$ and SQNR $=1.5\cdot 4^n$. Each extra bit adds about 6 dB. For a full-scale uniformly distributed signal ($\overline{m^2}=m_p^2/3$) the SQNR is $4^n$, i.e. $6.02n$ dB.

Minimum (ideal Nyquist, sinc pulses) baseband bandwidth for binary PCM; with a raised-cosine pulse of roll-off $\alpha$ it becomes $(1+\alpha )R_b/2$. The first-null bandwidth of rectangular NRZ pulses is $R_b$.

$\mu =255$ in the standard North American/Japanese system. In the limit $\mu \to 0$, $y=x$ (no compression).

For $m(t)=A\sin 2\pi f_{m}t$ this is $2\pi f_{m}A\le \Delta f_s$. If violated, the staircase cannot follow the signal (slope overload).

Equivalent to $p(kT)={\delta }_{k0}$ (normalised so $p(0)=1$). Smallest bandwidth satisfying it: $1/(2T)$ Hz, with $p(t)=\mathrm{sinc}(t/T)$, where $\mathrm{sinc}x=\sin (\pi x)/(\pi x)$ (ideal, not realisable).

Baseband (one-sided) bandwidth for symbol rate $R_s=1/T$. $\alpha$ is the roll-off factor. At $\alpha =0$ it is the ideal $R_s/2$; at $\alpha =1$ it is $R_s$.

For a baseband channel of bandwidth $B$. For a bandpass (double-sideband modulated) channel of bandwidth $B_c$ the limit is $R_s\le B_c/(1+\alpha )$.

With a raised-cosine pulse the passband bandwidth is $(1+\alpha )R_s$, so $\eta ={\log }_{2}M/(1+\alpha )$. Not for FSK, whose bandwidth also depends on the tone spacing.

The two tones must be orthogonal over one bit interval for the stated error probability to hold.

Output sampled at $t=T$. In the frequency domain $H(f)=S^{\ast }(f)e^{-j2\pi fT}$. The peak SNR depends only on the pulse energy, not its shape. A correlation receiver (multiply by $s(t)$ and integrate over $T$) gives the same output at the sampling instant $t=T$.

Sample $r=s_i+n$ with $s_1>s_0$ and noise $n\sim \mathcal{N}(0,{\sigma }^2)$ under both hypotheses. For antipodal levels $s_1=+a$, $s_0=-a$ it is $r_{th}=\frac{{\sigma }^2}{2a}\ln \frac{P(H_0)}{P(H_1)}$. For equal priors the threshold is the midpoint (ML). A less likely $H_1$ moves the threshold toward $s_1$.

Here $d^2=4E_b$ for BPSK. QPSK is two independent BPSK streams on the I and Q carriers, so its per-bit error probability is the same.

Orthogonal tones, $d^2=2E_b$. This needs 3 dB more $E_b/N_0$ than BPSK for the same BER.

Equal priors, threshold midway between the two received levels. Here $E_b$ is the average bit energy: the "1" has energy $2E_b$, the "0" none. If the same formula is written with the energy $E_1$ of the "1", use $Q(\sqrt{E_1/(2N_0)})$.

$S$ is the received signal power and $N=N_{0}B$ the noise power in bandwidth $B$. In dB: $(E_b/N_0{)}_{\text{dB}}={\text{SNR}}_{\text{dB}}-10{\log }_{10}(R_b/B)$.

Fraction of transmitted bits that carry message information; $0<R<1$ for an $(n,k)$ block code, since $n>k$.

A code with minimum distance $d_{min}$ is guaranteed to detect every pattern of up to $d_{min}-1$ bit errors, when it is used for detection only.

Guaranteed number of correctable errors when the receiver decodes to the nearest codeword (the spheres of radius $t$ around codewords do not overlap).

With this distance the decoder corrects every pattern of up to $t$ errors and, at the same time, detects (flags without miscorrecting) every pattern of $t+1$ up to $s$ errors. Here $s$ is a total number of errors, not extra errors.

Row vector $\mathbf{m}$ (length $k$) times $G$ gives the codeword $\mathbf{c}$ (length $n$), all operations modulo 2.

Message bits appear unchanged in the first $k$ positions of the codeword; $P$ is $k\times (n-k)$. Over GF(2), $-P^T=P^T$.

$H$ is $(n-k)\times n$. A word is a codeword exactly when every parity check is satisfied.

$\mathbf{s}=0$ means $\mathbf{r}$ is a valid codeword (no detectable error). A nonzero $\mathbf{s}$ means an error occurred.

Any binary $(n,k)$ code that corrects every pattern of up to $t$ errors must satisfy this: the $2^{n-k}$ syndromes must label every correctable error pattern, including the no-error pattern. A code that meets it with equality is called perfect.

Smallest $r$ that lets $m$ data bits be protected against any single error: the $r$ check bits must label the $m+r$ error positions plus the no-error case.

Full-length Hamming code, $r\ge 2$. Correct one error, or detect two (not both together). Since $1+n=2^r$, it meets the Hamming bound with $t=1$ exactly, so Hamming codes are perfect.

Probability that a block has more than $t$ errors. It is exactly the probability of not decoding the block correctly for a decoder that corrects every pattern of up to $t$ errors and no other (for example a Hamming code, $t=1$); for a decoder that also corrects some heavier patterns it is an upper bound.

For a linear code used for detection only: an error goes undetected exactly when the error pattern is itself a nonzero codeword. $A_i$ is the number of codewords of weight $i$.

Append $r$ zeros to the message, divide by $g(x)$ using XOR (no borrow), and the remainder $R(x)$ replaces the zeros. The transmitted word $T(x)$ is exactly divisible by $g(x)$.