A Mathematical Theory of Communication
1948 · paper · Computer science
Develops a general mathematical theory of communication, measuring information with a logarithmic measure. It extends earlier work to cover the effect of noise in the channel and the savings possible from the statistical structure of messages.
14 findings
Substance Section 16 says the Lagrange equations "∑_j p_sj log [p_sj / ∑_i P_i p_ij] = μ, s = 1, 2, …" determine the maximizing P_i and that "C will be the channel capacity". Requiring equality for every input s ignores the nonnegativity constraints, so the procedure fails when the optimum lies on the boundary (P_s = 0).
- p. 26 Section 16, Lagrange equations and their proposed solution
∑ⱼ pₛⱼ log[pₛⱼ/(∑ᵢ Pᵢpᵢⱼ)] = μ
Only the authors can say what is right here.
Judge's note: Checked the equations and the closing claim on the p.26 image. The paper imposes only ∑P_i = 1, plus the condition that the inverse h_st exists; it does not require P_i ≥ 0. I reworked the reviewer's invertible example (rows 0.9/0.05/0.05, 0.05/0.9/0.05, 0.49/0.49/0.02). At P = (1/2, 1/2, 0) the output distribution is (0.475, 0.475, 0.05). The row divergences are 0.4626 nats for rows 1 and 2 and 0.0121 nats for row 3, so the KKT conditions hold and this P is the optimum. The channel is invertible, so the optimal output distribution, and hence P, is unique. At that unique optimum the printed equal-divergence equations do not hold, so no valid probability vector solves them, and the printed procedure cannot return the capacity for this channel. The capacity definition itself is unaffected.
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Substance Appendix 3 states that with "H1 ≥ H2 ≥ ··· ≥ Hn", "Lim log n(q)/N = φ(q) is a decreasing step function, φ(q) = H_s in the interval ∑_1^{s−1} α_i < q < ∑_1^s α_i". But n(q) is non-decreasing in q, and the lowest-entropy component fills the smallest q, so the ordering is reversed.
- p. 30 Appendix 3, mixed-source theorem
H₁ ≥ H₂ ≥ ⋯ ≥ Hₙ
- p. 30 Appendix 3, mixed-source theorem
φ(q) is a decreasing step function
- p. 30 Appendix 3, theorem on mixed (non-ergodic) sources
is a decreasing step function
- p. 30 Appendix 3, ordering of component entropies
H1 ≥ H2 ≥ ··· ≥ Hn
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Judge's note: Checked p.30. n(q) counts the most probable sequences, taken in decreasing order of probability, until total probability q is reached. It is non-decreasing in q, so φ cannot decrease. Counterexample: a 1/2–1/2 mixture of a fair binary source (H1 = 1) and an all-zero source (H2 = 0). For every N the block 0^N has probability above 1/2, so n(1/4) = 1 and φ(1/4) = 0, whereas the printed theorem gives H1 = 1 on 0 < q < 1/2. Covering q = 3/4 needs about 2^(N−1) blocks, so φ = 1, whereas the printed theorem gives H2 = 0. Theorem 4 for ergodic sources is unaffected.
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Substance Section 21 (p. 38) prints "|log p / n − H′| < ε". Since H′ = −Lim (1/n)∫p log p, this should be |−log p / n − H′| < ε (log p^{−1}/n).
- p. 38 Section 21, last display
log p / n − H′ | < ε
Judge's note: The p. 38 image shows 'log p' over n with no minus sign, and H′ is defined on the same page with a leading minus. So (1/n)log p tends to −H′, and the printed inequality fails whenever H′ ≠ 0. The analogous Theorem 3 uses log p^{−1}/N, so the fix is evident.
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Substance Section 21 prints "|log p / n − H′| < ϵ", but H′ is defined just above as −Lim (1/n)∫p log p. It is −log p/n (equivalently log p/n + H′) that tends to H′.
- p. 38 Section 21, final displayed inequality
|(log p)/n − H′| < ε
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Judge's note: Checked the definition of H′ and the inequality at the foot of p.38, which continues on p.39. For white Gaussian noise, log p/n → −log√(2πeN) = −H′, so the printed absolute difference tends to 2H′, not 0. The discrete analogue on p.30, |log p/N − ∑P_i p_ij log p_ij| < η, has the consistent sign, since ∑P_i p_ij log p_ij = −H. The image shows no minus sign.
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Substance Theorem 14 gives "H2 = H1 + (1/W)∫_W log|Y(f)|² df" per degree of freedom. The per-degree-of-freedom change is (1/2W)∫ log|Y|² df, which is what the entropy-power factor on p.39, Table I and p.47 use.
- p. 39 Theorem 14
H2 = H1 + 1/W ∫W log|Y(f)|² df
- p. 39 End of Section 22
N1 exp 1/W ∫W log|Y(f)|² df
- p. 47 Lower bound for Theorem 20
W log4S − 2W = W log 4S/e²
- p. 39 Theorem 14
H₂ = H₁ + (1/W)∫ᵂ log|Y(f)|² df
Only the authors can say what is right here.
Judge's note: Checked p.39 (theorem, Jacobian, entropy-power factor), Table I on p.40 and p.47. H′ is defined per sample, and n = 2TW samples (p.38). The proof's Jacobian runs over n sine and n cosine coordinates, so log J ≈ T∫log|Y|² and the change per degree of freedom is (1/2W)∫log|Y|². Since entropy power is N = exp(2H′)/2πe, the paper's own factor exp((1/W)∫log|Y|²) corresponds to ΔH′ = (1/2W)∫log|Y|². For 1 − ω, (1/W)∫log(1−ω)² = −2, which gives Table I's 1/e². Under the theorem as printed the factor would be e^-4. On p.47 the per-second change ∫log G² = −2W also equals 2W × (1/2W)∫. Flat gain 2: the true change is log 2, the printed theorem gives log 4.
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Substance Appendix 7 defines "λ = Lim Lim Lim log N(ϵ,δ,T) / (T log ϵ)". Because log N ≥ 0 and log ϵ < 0 as ϵ → 0, this is never positive (−2W for a band-limited ensemble), contradicting the claim that it "agrees with the intuitive dimension rate" (2W, p.54).
- p. 55 Appendix 7, definition of λ
λ = Lim Lim Lim logN(ε,δ,T) / T log ε
- p. 54 Appendix 7
In the band limited case 2W numbers per second are sufficient.
- p. 55 Appendix 7, triple-limit definition
λ = Lim Lim Lim [log N(ε,δ,T)]/[T log ε]
- p. 55 Appendix 7, definition of dimension rate
T logε
Judge's note: Checked p.55: the denominator is T log ϵ with no minus sign. p.54 says '2W numbers per second are sufficient' in the band-limited case. Covering at scale ϵ needs N ≈ (c/ϵ)^{2WT}, so log N/(T log ϵ) → −2W. The fix is to use T log(1/ϵ) or add a minus sign.
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Wording In the first proof of Theorem 9 (pp. 16–17), the low-probability group is given "a total probability less than μ" and "η and μ approach zero". The rate bound "[(1−δ)T/N + δT1/N]^−1" and the line "As N increases δ, λ and φ approach zero" then use δ, which is never defined.
- p. 16 Proof of Theorem 9
having a total probability less than μ. As N increases η and μ approach zero
- p. 16 Rate formula, bottom of page
(1−δ)T/N + δ T1/N
- p. 17 Top line
As N increases δ, λ and φ approach zero
Only the authors can say what is right here.
Judge's note: On p. 16, μ is introduced as the bound on the probability of the second group. The displayed mean-rate formula at the bottom of p. 16 and the top line of p. 17 use δ in that role. The proof gives no definition of δ. It plausibly stands for the actual probability of the low-probability group (≤ μ), but only the authors can say whether δ or μ was meant.
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Wording Appendix 4 says "The correct value of D is the capacity C" and writes B_i = ∑B_j C^{−ℓ_ij}, |C^{−ℓ_ij} − δ_ij| = 0 and a rate "= C − …". The base of these powers must be W, where C = log W (Theorem 8, p.16). With base C the rate expression equals log C, not C.
- p. 31 Appendix 4
The correct value of D is the capacity C and the Bj are solutions of
- p. 16 Theorem 8
have a capacity C = logW. If we assign
- p. 31 Appendix 4, identification of D
The correct value of D is the capacity C
- p. 31 Appendix 4, final rate calculation
= C − [∑Pᵢpᵢⱼ log(Bⱼ/Bᵢ)]/[∑Pᵢpᵢⱼℓᵢⱼ]
Only the authors can say what is right here.
Judge's note: Compared p.31 with Theorem 8 on p.16 (p_ij = (B_j/B_i) W^{−ℓ}, C = log W) and Appendix 1 on p.28 (C = log W, D(W) = 0). Expanding −log[(B_j/B_i) C^{−ℓ_ij}] gives ℓ_ij log C − log(B_j/B_i), so the printed first equality '= C − …' holds only if the base is W. Check: one state with two unit-duration symbols. The printed equation B = B(2C^{−1}) forces C = 2, but the capacity is 1 bit. With W in place of C: W = 2 and C = 1. Theorem 8 as stated on p.16 is correct.
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Wording Appendix 3 (p. 30) defines the mixed source as "L = ∑ p_i L_i", but the theorem gives "φ(q) = H_s in the interval ∑_1^{s−1} α_i < q < ∑_1^s α_i". α_i is never defined.
- p. 30 Appendix 3, theorem for the mixed case
φ(q) = Hs in the interval
- p. 30 Appendix 3
L = ∑piLi
- p. 30 Appendix 3, interval for φ(q)
in the interval
Only the authors can say what is right here.
Judge's note: The image of p. 30 shows L = ∑p_iL_i, then the interval written with α_i. α_i appears nowhere else in Appendix 3. Presumably the mixture weights p_i are meant, in an order matched to the H_s, but the paper does not say so. The ordering also depends on C017.
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Wording Appendix 4 (p. 30): U includes "∑ η_j P_i(p_ij − δ_ij)", but the derivative is printed "∂U/∂p_ij = −[MP_i(1+log p_ij)+NP_iℓ_ij]/M² + λ + μ_i + η_i P_i = 0". The constraint term gives η_j P_i, not η_i P_i.
- p. 30 Appendix 4, ∂U/∂p_ij
+λ+µi +ηiPi = 0
Only the authors can say what is right here.
Judge's note: The image of p. 30, and the text layer ("+∑ jPi(pij" in U, "iPi" in ∂U), show η_j in U and η_i in the derivative. My own working: ∂/∂p_ij of ∑_{k,l} η_l P_k(p_kl − δ_kl) = η_j P_i. With η_i printed, solving for log p_ij leaves only i-dependent terms apart from ℓ_ij, so the factor B_j in p_ij = A_iB_jD^(−ℓ_ij) on p. 31 could not arise. With η_j it does. The step is faulty as printed, but the final result (Theorem 8) stands. The λ term is also spurious, since λ∑P_i does not depend on p_ij, but this is harmless.
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Wording Appendix 4 (p. 30): "Evidently for a maximum p_ij^(s) = k exp ℓ_ij^(s)". As printed, the probability grows with symbol duration. Both Theorem 8 (p. 16: p_ij^(s) = (B_j/B_i)W^(−ℓ_ij^(s))) and the appendix's own result p_ij = A_iB_jD^(−ℓ_ij) (p. 31) make it fall with duration.
- p. 30 Appendix 4, first paragraph after the rate expression
Evidently for a maximum p(s)ij = k expℓ(s)ij
Only the authors can say what is right here.
Judge's note: The image of p. 30 shows "k exp ℓ_ij^(s)" with no minus sign and no rate constant in the exponent. Check with two symbols of durations 1 and 2, all sequences allowed: W solves W^−1 + W^−2 = 1, so W ≈ 1.618 and the optimal probabilities are ≈ 0.618 and 0.382, decreasing in ℓ. The printed form gives a ratio p2/p1 = e > 1. The step is faulty as printed, but the stated result (Theorem 8) stands.
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Wording Section 20, item 8 (p. 37): "H(y) = ∫···∫ p(x1,...,xn)J(x/y) log p(x1,...,xn)J(x/y) dy1···dyn" has no leading minus sign.
- p. 37 Section 20, property 8, first display
H(y) = ∫···∫ p(x1,...,xn)J
Only the authors can say what is right here.
Judge's note: On the p. 37 image the integral follows '=' directly, with no minus. The entropy definition on the same page reads H(x) = −∫p log p dx. Take q(y) = pJ. Then H(y) = −∫ q log q dy = −∫ pJ log(pJ) dy, and changing variables gives H(x) − ∫p log J dx, which is the next display. As printed, the identity map (J = 1) gives H(y) = −H(x), contradicting H(y) = H(x) on the next line. The final result stands.
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Wording Appendix 6 normalizes the Gaussians as "p(xi) = |Aij|^{n/2}/(2π)^{n/2} exp −½∑Aij xi xj" (likewise for q, and "log r = n/2 log (1/2π)|Cij| …"). The exponent should be 1/2, as Section 20 item 6 on p.37 has ("|aij|^{1/2}/(2π)^{n/2}").
- p. 53 Appendix 6, normal densities
p(xi) = |Aij|^{n/2} / (2π)^{n/2} exp −½∑Aijxixj
- p. 37 Section 20, item 6
|aij|^{1/2} / (2π)^{n/2} exp(−½∑aijxixj)
- p. 53 Appendix 6, definitions of normal p and q
p(xᵢ) = |Aᵢⱼ|^(n/2)/(2π)^(n/2) exp[−(1/2)∑Aᵢⱼxᵢxⱼ]
- p. 37 Section 20, property 6, general Gaussian density
|aᵢⱼ|^(1/2)/(2π)^(n/2)
Only the authors can say what is right here.
Judge's note: Compared p.53 with p.37. ∫exp(−½xᵀAx)dx = (2π)^{n/2}|A|^{−1/2}, so the printed density integrates to |A|^{(n−1)/2}. For n = 2 and A = 2I that is 2, not 1. The log r line carries the same factor. The argument matches only the quadratic terms, so the conclusion stands.
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Wording Appendix 6 prints "H1∫q(xi − si) log r(xi) dxi = −H3 log p(si)" (and likewise "= −H3 log q(si)"). From the preceding "H3 = −λH1" the substitution gives +H3 log p(si).
- p. 53 Appendix 6
H1 ∫ q(xi − si) log r(xi) dxi = −H3 log p(si)
- p. 53 Appendix 6
H3 = −λH1
- p. 53 Appendix 6, after 'solving for λ and μ and replacing in the equations'
= −H3 log p(si)
Only the authors can say what is right here.
Judge's note: Checked p.53. Multiplying ∫q log r = −λ log p by p(s) and integrating gives −H3 = λH1, so H3 = −λH1 is printed correctly. Then −λ = H3/H1, so ∫q log r = (H3/H1) log p, a positive coefficient. The page's own Gaussian check ('This should equal (H3/H1)[…]') uses the + sign, so the minus sign in the intermediate pair is a slip. The final conclusion A_ij = (H1/H2)B_ij is unaffected.
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25 minor (typos and the like; most checked by the judge only)
Wording P. 19: "All numbers up to 16 are represented as usual; 16 is represented by the next binary number after 16 which does not contain four zeros, namely 17 = 10001". 16 = 10000 is the first number containing 0000, so the numbers kept "as usual" end at 15.
- p. 19 Section 10 Discussion and Examples, second example (A's and B's), coding paragraph
All numbers up to 16 are represented as usual; 16 is represented by the next binary number after 16
- p. 19
All numbers up to 16 are represented as usual; 16 is represented by the next binary number
Judge's note: The image of p. 19 matches the extracted text. 16 in binary is 10000, which contains the special sequence 0000, so it is the first number that is replaced. "Up to 16" works only if "up to" excludes 16; read inclusively, the sentence contradicts itself. This is a wording ambiguity, not a computational error.
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Wording Appendix 1 asserts that "the same growth properties result if we require that all blocks start in the same (arbitrarily chosen) state". This fails when the state graph is not strongly connected, and no such condition is stated.
- p. 28 Appendix 1, final assertion
the same growth properties result if we require that all blocks start in the same (arbitrarily chosen) state
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Judge's note: Checked p.28. Neither Section 1 nor Appendix 1 states an irreducibility condition. Reviewer's counterexample: state A has one unit self-loop, state B has two, and the states are not connected. D(W) = (W^-1 − 1)(2W^-1 − 1) has largest root W = 2, so C = log W = 1 bit. Blocks that start in A number 1 for every length L, a growth rate of 0. The claim therefore fails for an arbitrarily chosen start state. C = log W itself still holds, so the step is faulty but the stated result stands.
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Wording Appendix 4 asserts "Evidently for a maximum p_ij^(s) = k exp ℓ_ij^(s)". As printed, this makes the probabilities grow with symbol duration; at the maximum they decrease with duration (∝ W^{−ℓ}).
- p. 30 Appendix 4, immediately before the constrained maximization
Evidently for a maximum pᵢⱼ⁽ˢ⁾ = k exp ℓᵢⱼ⁽ˢ⁾.
Only the authors can say what is right here.
Judge's note: Checked p.30: the exponent has no minus sign and no multiplier. Reviewer's example: one state, two symbols of durations 1 and 2. The printed form forces p2/p1 = e, giving p = (0.2689, 0.7311) and about 0.485 bits per unit time. The feasible choice (0.618, 0.382) gives about 0.694, so the printed form is not the maximum. The later steps (p.31: p_ij = A_i B_j D^{−ℓ_ij}) and Theorem 8 use the decreasing form, so the stated result stands.
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Wording Theorem 13 ("Let f(t) contain no frequencies over W", X_n = f(n/2W)) includes the band-edge frequency W. f(t) = sin 2πWt has all samples equal to zero, so the printed series reconstructs 0.
- p. 34 Theorem 13
Let f(t) contain no frequencies over W.
- p. 34 Theorem 13, sample definition
Xₙ = f(n/(2W))
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Judge's note: Checked p.34. The hypothesis "no frequencies over W" admits frequency exactly W, and the theorem states no finite-energy condition; the proof is deferred to the cited 1949 IRE paper, which is not among the files. sin(2πW·n/2W) = sin(πn) = 0 for every n, but f(1/4W) = 1. This is a boundary-case gap in the statement, not a problem for typical band-limited ensembles.
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Wording p. 47: "the output is peak limited to ±S". In the same paragraph S is the peak power and the sample amplitudes are ±√S.
- p. 47 Section 26, small-S/N argument
the output is peak limited to ±S
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Judge's note: The p. 47 image shows '±S'. A few lines earlier the samples are '+√S or −√S'. With a unit-DC-gain triangular filter the amplitude bound is ±√S, unless 'peak power S' was meant. This is a loose notation slip, so it is minor.
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Wording The Section 29 Lagrangian (p. 50) uses multiplier μ on the ρ term ("+ μP(x,y)ρ(x,y) + ν(x)P(x,y)"), but the solution (p. 51) is written "P_y(x) = B(x)e^{−λρ(x,y)}" with no statement relating λ to μ.
- p. 50 Section 29, Lagrangian
+ μP(x,y)ρ(x,y) + ν(x)P(x,y)
- p. 51 Section 29, solution
where λ is determined to give the required fidelity
Judge's note: Checked the p. 50 and p. 51 images. On p. 50, λ is already used for the constraint function λ(x,y), so the switch is a notational inconsistency. The result is unaffected.
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Wording Section 29: from "∫B(x)e^{−λρ(x−y)}dx = 1" the paper concludes "Hence B(x) is constant". A constant convolution does not force B to be constant when the kernel's Fourier transform has zeros.
- p. 51 Section 29, translation-invariant distance calculation
∫ B(x)e^(−λρ(x−y)) dx = 1.
- p. 51 Following sentence
Hence B(x) is constant
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Judge's note: Checked p.51. Reworked the reviewer's example: k(z) = [φ(z−π/2) + φ(z+π/2)]/2 with ρ = −log k and λ = 1 (ρ > 0 because k < 1). ∫k(z)cos z dz = e^{−1/2}cos(π/2) = 0, and ∫k(z)sin z dz = 0 by symmetry, so B(x) = 1 + ½cos x gives ∫B(x)k(x−y)dx = 1 for every y, yet B is not constant. The inference as printed does not follow. The squared-error Gaussian case used later is unaffected.
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Wording Theorem 22 "R = W1 log Q/N" and Theorem 23 "W1 log Q1/N ≤ R ≤ W1 log Q/N" carry no restriction N ≤ Q. For N > Q they give a negative rate, whereas the rate is 0 (an independent reproduction achieves the allowed error).
- p. 51 Theorem 22
R = W₁ log(Q/N)
- p. 51 Theorem 23
W₁ log(Q₁/N) ≤ R ≤ W₁ log(Q/N)
Only the authors can say what is right here.
Judge's note: Checked p.51. Neither theorem restricts N, although the proof's sphere of radius √(Q−N) implicitly assumes N ≤ Q. Example: Q = 1, N = 2, with an independent reproduction of power 1. Then E(x−y)² = 2 and the mutual information is 0, but the printed formula gives log(1/2) < 0. This is a missing boundary condition, not an error in the main case.
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Wording Appendix 6 gives the conditions for a minimum as "∫q(xi − si) log r(xi) dxi = −λ log p(si)" without the additive constants that arise from the normalization constraints ∫p = ∫q = 1, which U does not include.
- p. 53 Appendix 6, asserted conditions for a minimum
∫q(xᵢ−sᵢ) log r(xᵢ) dxᵢ = −λ log p(sᵢ)
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Judge's note: Checked p.53: U contains multipliers only for the entropy constraints, not for normalization. Example: p = q = N(0,1), r = N(0,2), so L(s) = −½log 4π − (s²+1)/4. Matching the s² terms forces λ = −1/2, but a constant of about −1.056 then remains, so the stationary Gaussian pair fails the printed condition. The paper's own Gaussian check matches only the quadratic terms. The variational step needs normalization constants. This is a gap in the proof, not a counterexample to the stated inequality.
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Wording Appendix 7 defines "R1 = (1/T)∑_i P(Xi,Yi) log [P(Xi,Yi)/(P(Xi)P(Yi))]" as a single sum over matching indices. As written it omits the off-diagonal intersections Xi∩Yj, and its value depends on how the Y cells are labelled.
- p. 54 Appendix 7, definition of R₁
R₁ = (1/T)∑ᵢ P(Xᵢ,Yᵢ) log[P(Xᵢ,Yᵢ)/(P(Xᵢ)P(Yᵢ))]
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Judge's note: Checked p.54. X_i and Y_i are introduced as separate partitions, but the sum has one index. The refinement argument refers to 'the X1, Y1 intersection', which suggests a sum over all intersections was intended, so this is mainly a notational ambiguity. Example: equiprobable binary X with Y = 1−X. The diagonal terms are zero, so the printed sum is 0 bits, while the double sum gives 1 bit.
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Typo Subject–verb agreement error on p. 1: "since we in-tuitively measures entities by linear comparison" should read "measure".
- p. 1 Introduction, reason 2
since we in-tuitively measures entities by linear comparison
- p. 1 Introduction, item 2 of the logarithmic-measure list
we in-tuitively measures entities by linear comparison
- p. 1
tuitively measures entities by linear comparison with common standards
Judge's note: The crop of p. 1, item 2, shows "we in- tuitively measures entities". The hyphen is just the line break; "measures" is really printed.
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Typo Typo on p. 2, item 1(a): "A sequence of letters as in a telegraph of teletype system" should read "telegraph or teletype system".
- p. 2 Item 1(a), information source
A sequence of letters as in a telegraph of teletype system
- p. 2 Introduction, item 1(a) describing message types
as in a telegraph of teletype system
- p. 2 Communication system, item 1(a)
in a telegraph of teletype system
Judge's note: The crop of p. 2 shows "as in a telegraph of teletype system". Read as "of", the phrase makes no sense; "or" is evidently meant.
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Typo The same quantity, the duration of the s-th symbol from state i to state j, is written b_ij^(s) in Theorem 1 (p. 4) and Appendix 1, but ℓ_ij^(s) in Theorem 8 (p. 16) and Appendix 4.
- p. 4 Theorem 1
Let b(s)ij be the duration of the sth symbol
- p. 16 Theorem 8
is the duration of the sth symbol leading from state i to state j
Judge's note: On p. 4 Theorem 1 says "Let b_ij^(s) be the duration of the s^th symbol". On p. 16 Theorem 8 says "where ℓ_ij^(s) is the duration of the s^th symbol leading from state i to state j". Each symbol is defined where it is used, so the meaning can be recovered; this is cosmetic only.
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Typo A sentence on p. 16 starts with a lowercase letter: "small when T is large. if we choose" should read "If we choose".
- p. 16 Proof of Theorem 9, first method
small when T is large. if we choose
Judge's note: The crop of p. 16 shows a full stop followed by lowercase "if".
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Typo On p. 19 the joint entropy is written "The joint entropy of input and output will be H(xy)." A few lines later, and elsewhere in the paper, it is written H(x,y).
- p. 19 Section 11, last paragraph
The joint entropy of input and output will be H(xy).
Judge's note: The image of p. 19 shows "H(xy)" in the sentence and "H(x,y) = H(x)+H_x(y)" in the next display. The meaning is clear; this is a cosmetic notation inconsistency.
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Typo Duplicated word on p. 24: "If the average of a set of numbers is within ε of of their maximum".
- p. 24
of a set of numbers is within of of their maximum, a fr
Judge's note: The crop of p. 24 shows "within ε of of their maximum". The text layer drops the ε, but the doubled "of" is really printed.
Second check: code check
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Typo Number agreement in Appendix 1 (p. 28): "where b^1_ij, b^2_ij, ..., b^m_ij are the length of the symbols" should read "lengths".
- p. 28 Appendix 1
are the length of the symbols which may be chosen in state i
Judge's note: The crop of p. 28 shows "are the length of the symbols which may be chosen in state i". The subject is plural, so the noun should be "lengths". Grammatical slip only.
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Typo Appendix 4 (p. 31): "So that if λ_i satisfy" is followed by equations in γ_i (∑γ_iC^(−ℓ_ij) = γ_j, P_i = B_iγ_i) and by "equations for B_i and γ_i". λ_i should be γ_i.
- p. 31 Appendix 4
So that if λi satisfy
- p. 31 Appendix 4
Pi = Biγi
Judge's note: The image of p. 31 shows "So that if λ_i satisfy", followed by displays and text that use only γ_i. λ is already used on p. 30 for a different Lagrange multiplier. This is an evident symbol slip.
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Typo Footnote 3, p. 33: "by Birkoff, von Neumann, and Koopman". The mathematician's name is Birkhoff.
- p. 33 Footnote 3
by Birkoff, von Neumann, and Koopman
Only the authors can say what is right here.
Judge's note: Rests on outside knowledge: the mathematician G. D. Birkhoff spells his name 'Birkhoff'. The crop shows 'Birkoff' as printed.
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Typo p. 34: "It is easily shown (see Appendix 5 that if T is invariant". The closing parenthesis is missing.
- p. 34 Section 18
It is easily shown (see Appendix 5 that if T is invariant
Judge's note: The crop shows the parenthesis opened before 'see' and never closed in the sentence.
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Typo p. 39: "This follows form the ergodic property". 'form' should be 'from'.
- p. 39 Top paragraph
This follows form the ergodic property
- p. 39
This follows form the ergodic property if we divide
Judge's note: Seen in the crop.
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Typo p. 40: "any gain characteristic obtain from 1 − ω". This should read 'obtained from'.
- p. 40 Paragraph after Table I
any gain characteristic obtain from 1 − ω
Judge's note: Seen on the p. 40 image, in the paragraph after Table I.
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Typo p. 44: "in terms of the average noise power N the noise entropy power N1". The word 'and' is missing between the two quantities.
- p. 44 Paragraph before Theorem 18
in terms of the average noise power N the noise entropy power N1
Judge's note: Seen on the p. 44 image. N and N1 are different quantities (Theorem 18 defines both), so this cannot be read as apposition.
Second check: none (judge only)
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Typo p. 44: "since we have shown in in a previous theorem". The word 'in' is duplicated.
- p. 44 Proof of Theorem 18
since we have shown in in a previous theorem
- p. 44 Section 25, lower-bound derivation
since we have shown in in a previous
- p. 44
er P + N1 since we have shown in in a previous theorem
Judge's note: Seen on the p. 44 image.
Second check: code check
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Typo Appendix 7 (p. 54): "non-overlapping measurable subsets Xi and Yi approximate to the rate of transmission R by". A word is missing (e.g. 'and' before 'approximate'), so the sentence does not parse.
- p. 54 Appendix 7, first paragraph
non-overlapping measurable subsets Xi and Yi approximate to the rate of transmission R by
Judge's note: Seen in the crop.
Second check: none (judge only)