Shannon, C.: Communication in the presence of noise. Proc. Inst. of Radio Engineers 37 (1949) 10--21  Home/Search   Document Not in Database   Summary   Related Articles   Check

....a signal is possible either in the time domain t or the frequency domain w, but not both simultaneously. This is due to the fact that the Fourier transform is computed using e iwt : Since the product wt must remain constant, narrowing a function in one domain, causes it to be wider in the other [7, 12]. For example, a pure sinusoidal wave has no time resolution, as it possesses nonzero components over the in nitely long time axis. Its Fourier transform, on the other hand, has excellent frequency resolution: It is an impulse function with a single positive frequency component. By contrast, an ....

C.E. Shannon, Communication in the presence of noise, Proceedings of the IRE, Vol. 37, 1949, 10-21.

....and a Schmmitt trigger. We show how to recover the amplitude information of a bandlimited signal from the time sequence loss free. 1. INTRODUCTION A fundamental question arising in information processing is how to represent a signal as a discrete sequence. The classical sampling theorem ( 6] [10]) calls for representing a bandlimited signal based on its samples taken at or above the Nyquist rate. X(#) 0 for # # x(t) z(t) x(t) z(t) t b x(t) Time encoding machine (TEM) Time decoding machine (TDM) t 2 t 0 t 3 t 1 Fig. 1. Time Encoding and Decoding. A time encoding of a ....

....bandlimited assumption on x to obtain a perfect reconstruction of the signal even though the trigger times are irregular. As expected, the interval between two consecutive trigger times has to be smaller then the distance between the uniformly spaced samples in the classical sampling theorem [6] [10]. The mathematical methodology used here is based on the theory of frames [3] We shall construct a linear operator on L , the space of square integrable functions defined on R, and by starting from a good initial guess followed by successive interations, obtain sucessive approximations that ....

Shannon, C.E., Communications in the Presence of Noise, Proceedings of the IRE, Vol. 37, pp. 10-21, January 1949. 5

....processing, mathematical uncertainty principles and learning theory, among other areas. Signal Processing. There are many signal processing results pertaining to signal recovery from sampled values of a function. We restrict ourselves to the classical result: the Shannon sampling theorem [14]. It says that you can exactly reconstruct a continuous time function F from its sampled values, provided the nonzero Fourier coefficients all lie within the interval [ Gamma=T; T ] and the samples are taken at regular spaced points in time, each separated by T . If the largest frequency that F ....

C. E. Shannon. Communications in the presence of noise. Proc. of the IRE (37), 10--21, January 1949.

....thus providing a trade o# between the sometimes excessive sharpness of the Catmull Rom and the blurriness of the B spline. C. Frequency Domain Analysis A frequency domain analysis compares the reconstructors against the sinc function, the perfect reconstructor for band limited signals [18] [19]. While this is accepted practice, it has been noted that it does not correlate well with subjective visual analysis [8] 9] 11] Figures 5 and 6 show the frequency responses of various reconstructors. Only the positive frequency axis is shown as all of the functions are even. Two things are ....

C. E. Shannon, "Communication in the presence of noise", Proc. IRE, Vol. 37, No. 1, Jan, 1949, pp.10--21.

....does not have polynomial size circuits would imply that P 6= NP. Largely because of many such implications for complexity classes, considerable e ort has been devoted to proving circuit lower bounds. However, to this point this e ort has met with limited success. In an early paper, Shannon [Sha49] showed that most Boolean functions require exponential size circuits. This proof was nonconstructive and proving bounds on particular functions is more dicult. In fact, no non linear lower bound is known for the circuit size of a concrete function. To get more positive results one needs to ....

C.E. Shannon. Communication in the presence of noise. IRE, 37:10-21, 1949.

....1. DATA HIDING AS ADDITIVE NOISE 1.1. Motivation The motivation to model the steganographic process as the addition of noise arises from a number of factors. In the process of sampling and transmitting signals there are numerous sources of noise such as quantization[1] sensor[2] and channel[3]. A number of steganographic hiding schemes have used this as a foundation for noise based data hiding. The goal is to disguise the message as a naturally present noise and add it to the coverimage. While the additive noise framework is especially well suited to schemes which rely on noise based ....

C. E. Shannon, "Communication in the presence of noise," Proceedings of the I.R.E. 37, pp. 10--21, Jan. 1949.

....3G mobile communication systems, also known as Universal Mobile Telecommunications System (UMTS) Taking into account that the frequency band of 3G mobile communication systems lies around 2. 5 GHz, we conclude that we need ADCs with sampling rate at least 5 GHz, according to the sampling theorem [4]. In practical SDR solutions, ADCs are placed after the first down conversion stage thus significantly reducing the required sampling rates [5] The latest technology in silicon integrated circuits (Si ICs) ADCs has reached such levels of sampling rates that can handle band limited signals having ....

....OF THE BASIC CHARACTERISTICS OF AN ADC A. Sampling Rate According to the sampling theorem, every band limited signal of f mite energy, having no spectral components above a frequency of fc, may be reproduced from its samples taken at a rate exceeding twice the highest frequency in the band [4]. The sampling rate fs=2fc is called the Nyquist rate (fNyquist) The sampled signal xs(t) is given by the relation: x, t) x(t) y 7(t nT, 1) where x(t) is the analog signal, 5(0 stands for Dirac delta function, T is the sampling period and is equal to 1 f. The Fourier Wansform X(f) of the ....

C.E. Shannon, "Communication in the Presence of Noise", in Proceedings of IRE, 1949, vol. 37 pp. 10-21.

....to reconstruction, is also described. 2.4.1 Data Interpolation be a continuous 1D function. A discrete approximation to , notated , is constructed by sampling the function at intervals. Here, we assume that these intervals are regular. According to Shannon s Sampling Theorem [Sha49] a continuous function with a maximum frequency of Hz can be reconstructed from a discrete function with total accuracy, but only if the discrete function has a frequency of at least (the Nyquist rate) If is sampled at or above the Nyquist rate therefore, then it is possible to ....

....of accuracy, the ability to compute normal vectors for semi transparent volumes, and viewpoint independence. A normal vector at a point in a discrete volume, is the partial derivative of with respect to the , and axes: According to Shannon s Sampling Theorem [Sha49] can be computed precisely (assuming that has a frequency above the Nyquist rate [BLM96] by convolving with the derivative of , shown in Equation 2.4. However, as with the filter described earlier, the derivative, has an infinite extent and must therefore be truncated ....

C.E. Shannon. Communication in the presence of noise. Proceedings of the IRE, 37:10--21, January 1949.

....most fundamental and importantpieceofworkinthisareabeingtheclassicalsamplingtheorem. AlsoknownastheWhittaker Koteln ikov Shannon(WKS) theorem, it states that a lowpasssignalbandlimited to thefrequencies can be reconstructed perfectly from its samples takenuniformlyatnolessthantheNyquistrateof [5].Another importantresultinsamplingtheoryduetoLandauisa lowerbound on the samplingdensityrequired for anysamplingscheme that allowsperfectreconstruction [6] Formultiband signals, this fundamentallowerboundisgivenbythetotallength (measure)ofsupport of the Fourier transform of the signal. Landau s ....

C. E. Shannon, "Communication in the presence of noise," in Proc. IRE, vol. 37, Jan. 1949, pp. 10--21.

....and additive (quantization) noise. In addition, it is not unreasonable to expect that an attacker might use filtering (because of its simple implementation) or Gaussian noise (since in an additive noise chanel with limited noise variance, communication is most difficult when the noise is Gaussian [40]) Finally, just as a real world original signal may be modeled as being locally stationary, the attack model can represent locally stationary processing of the watermarked signal. Recall that the frequency supports of the watermark and original are , respectively. Clearly should be a ....

....y###. The result is just like an AWGN channel with power constraint # # and noise power # # , and the capacity is # # # # [39, 14] If w#### and v#### are # D with respective power spectra ### #### and # ## ####, then the capacity is # ##### ## ### # ## # ### ###### ## ##### ### [40]. In blind watermarking, x### is known to the encoder but not to the decoder. This scenario was considered by Costa [13] who proved the remarkable and surprising result that the capacity is again # # ### # ## ## # #. Hence, blind watermarking can theoretically perform as well as ....

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C. E. Shannon, "Communication in the presence of noise," Proc. IRE, Vol. 37, pp. 10--21, 1949.

....numerical analysis, and signal and image processing. Successful interpolants include polynomials, harmonic waves, radially symmetric functions, nite elements, splines, wavelets, etc. Despite the diversity of the literature, there indeed exists one most widely recognized result due to Shannon [50], known as Shannon s Sampling Theorem. Theorem 1 (Shannon s Theorem) If a signal u(t) is bandlimited within ( then, u(t) n=1 sinc t n 7 That is, if an analog signal u(t) with nite energy, or equivalently, in L (IR ) does not contain any high frequencies, then ....

C. E. Shannon. Communications in the presence of noise. Proc. I.R.E., 37:10-21, 1949.

....component algorithms discussed above were concerned with optimizing information given noise on the input to the network. These decorrelating algorithms instead attempt to optimize information in the case of noise on the output of the network. To analyze this, we follow an argument by Shannon [22]. Consider a network with input represented by the random variable X , output Y , with added noise giving a nal information bearing output = Y . For small output noise, we can express the transmitted information as I( X) 1=2 log det C Y 1=2 log det C (4.24) and the power cost as S T ....

C. E. Shannon. Communication in the presence of noise. Proceedings of the IRE, 37:10-21, 1949.

....within a given abscissa range. Here, we have chosen to make use of a separable model because such a model is easy to manipulate. It is given by f (xx) c(kk) j(x i k i ) q kkZ , 27) where c(kk) is a discrete set of coefficients and j(x) is a continuous synthesis function. For example, [45] has proposed to use j(x) sinc(x) This choice limits f (xx) to being a finite bandwidth function with coefficients given by c(kk) f (kk) We prefer to use j(x) b (x) where b is a symmetrical spline of degree n . In this case, f (xx) is a piece wise polynomial function, and the ....

C.E. Shannon, "Communication in the Presence of Noise," Proceedings of the I. R . E., vol. 37, pp. 10--21, January 1949.

.... such as rotations and scaling (interpolation) The textbook approach to those problems is provided by Shannon s sampling theory which describes an equivalence between a bandlimited function and its equidistant samples taken at a frequency that is superior or equal to the Nyquist rate [76]. Even though this theory has had an enormous impact on the field, it has a number of problems associated with it. First, it relies on the use of ideal filters which are devices not commonly found in nature. Second, the bandlimited hypothesis is in contradiction with the idea of a finite (or ....

C.E. Shannon, "Communication in the presence of noise," Proc. I.R.E., vol. 37, pp. 10-21, 1949.

....papel; Index Terms Reconstruction, sampling, thin plate splines, vari ational criterion. I. INTRODUCTION A. Sampling and Reconstruction ECONSTRUCTING a signal iom its samples is one of the most fundamental tasks in signal processing. The classical sampling theorem presented by Shannon [2] states that a bandlimited function fi, whose frequency spectrum is limited by the Nyquist frequency o: w T) can be re constructed perfectly from its regularly spaced (ideal) samples 8; fi(jT) by convolution with a sinc kernel = y 8; sinc(x T j) where sinc(x) sin(7rx) 71 37 (i) ....

C.E. Shannon, "Communication in the presence of noise," Proc. IRE, vol. 37, pp. 10-21, Jan. 1949.

....to the traditional criterion, is meaningful for finite bandwidth communication. Index Terms Antenna arrays, channel capacity, fading channels, noisy channels, spectral efficiency, wideband regime. I. INTRODUCTION S HORTLY after A Mathematical Theory of Communication, Claude Shannon [1] pointed out that as the bandwidth tends to infinity, the channel capacity of an ideal bandlimited additive white Gaussian noise (AWGN) channel approaches (b s) 1) where is the received power and is the one sided noise spectral level. Since capacity is monotonically increasing with bandwidth , ....

C. E. Shannon, "Communication in the presence of noise," Proc. IRE, vol. 37, pp. 10--21, Jan. 1949.

....development of the error as T 0, and also sharp (asymptotically exact) upper bounds. I. Introduction Re sampling and interpolation play a central role in image processing. These operations are required to rescale, rotate images or to correct for spatial distortions. Shannon s theory [1] provides an exact sampling interpolation system for bandlimited signals. However, this method is rarely used in practice because of the slow decay of sinc(x) Instead, practitioneers rely on more localized methods such as bilinear interpolation, short kernel convolution and polynomial spline ....

C.E. Shannon, "Communication in the presence of noise," in Proc. IRE, January 1949, vol. 37, pp. 10--21.

.... it is near optimal for small D (or large number of bins) assuming A is well behaved (for example, when A is Laplacian [47] or the quantizer and A satisfy the high bit rate assumption described in [21] Note that the entropy of A is a lower bound for the average bit size needed to code A [34]. In the context of progressive transmission, there are two types of scalar quantization: multiscale and embedded quantization. In multiscale quantization, the data is first quantized by a coarse quantizer and the quantization error is then quantized by a finer quantizer. In the embedded ....

C.E. Shannon. Communications in the presence of noise. In Proceedings of the IRE, volume 37, pages 10--21, January 1949. 109

....performed through interpolation consists in sampling data values of the underlying continuous function at the right set of abscissa. Additional hypotheses have been proposed in the past; for example, a traditional approach is to restrict the underlying continuous function to be bandlimited [5]. More recent approaches suggest that replacing this latter hypothesis by an alternative one allows better computational efficiency, while keeping the validity of the two fundamental hypothesis expressed above. More specifically, dealing with spline functions offers an alternative to bandlimited ....

C.E. Shannon, "Communication in the Presence of Noise," Proceedings of the I. R . E., vol. 37, pp. 10--21, January 1949.

.... coding The original justi cation for the redundancy reduction hypothesis considered for example signals with a nite number of levels, such as just noticeable di erences (JNDs) However, a related approach was formulated by Shannon for economical transmission of signals through a noisy channel [41]. Consider a system (a neural network in this case) which is transmitting its real valued output signal Y through a channel where it will be corrupted by additive noise (Fig. 4) If there were no restrictions on Y , we could simply amplify it until we had overcome as much of the noise as we ....

....to maximise J = I( X) S T = S(f) N(f) N(f) S(f) as a function of S(f) for every 0 f B. This is the case when S(f) N(f) constant (17) so if N(f) is white noise, i.e. the power spectral density is uniform (or at) the power spectral density S(f) should also be uniform [41]. A lter which performs this attening is called a whitening lter. It is well known that a signal with at power spectral density has an autocorrelation function R y y( E (Y (t) Y (t ) which is proportional to a delta function ( In other words, the time varying output signal Y (t 1 ....

C. E. Shannon. Communication in the presence of noise. Proceedings of the IRE, 37:10-21, 1949.

....errors, that arise as a consequence of discretization and reconstruction. 10.4.1 Aliasing Errors Sampling Theorems What we normally think of as the sampling theorem is the theorem named after E. T. Whittaker, J. M. Whittaker, V. A. Kotelnikov or C. E. Shannon [Whi15, Whi29, Whi35] Kot33] [Sha49], cited here after [Jer77] It states, that a band limited function can be reconstructed exactly from regular samples of the function, if these samples are close enough (at least two samples per smallest involved wavelength) It is less known however, that generalizations of this WKS sampling ....

C. E. Shannon. Communications in the presence of noise. Proc. IRE, 37:10--21, January 1949.

....using orthogonal (and orthonormal) weight vectors developed or at least, this problem of avoiding potential degeneracy may help to explain why the convention of variance preservation has not been seriously challenged. 2. 3 Dealing with output noise Using an approach originally taken by Shannon [20], we earlier considered the problem of adapting a linear network to maximize transmitted information, but with limited power cost. The optimum corresponds to a network whose outputs are decorrelated and have equal variance [15] This can be achieved using a network with symmetrical lateral ....

C. E. Shannon. Communication in the presence of noise. Proceedings of the IRE, 37:10--21, 1949.

....additive (quantization) noise. In addition, it is not unreasonable to expect that an attacker might use filtering (because of its simple implementation) or Gaussian noise (since in an additive noise chanel with limited noise variance, commu nication is most difficult when the noise is Gaussian [40]) Finally, just as a real world original signal may be modeled as being locally stationary, the attack model can represent locally stationary processing of the watermarked signal. Recall that the frequency supports of the watermark and original are 142 and A , respectively. Clearly 142 should be ....

....on Costa s solution. To extend Costa s result to an M D Gaussian channel with memory and channel state known to the encoder, one can divide the frequency spectrum into pallel, independent Gaussian subchannels, apply the result to each subchannel, and let the number of subchannels go to infinity [40]. Then for fixed power spectra ( and ( C = 2= fn log= 1 ( d. Our water.king model includes filtering and the original interference suppression factor a. Conse quently, we use (3) to write the capacity [29] as We may interpret (7) as follows. We say that the effec6ve ....

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C.E. Shannon, "Communication in the presence of noise," Proc. IRE, Vol. 37, pp. 10-21, 1949.

....Y ) Gamma S (4) where is a Lagrange multiplier. For a Gaussian signal Y with spatially invariant statistics (much as we might expect in vision) and white Gaussian noise Phi, J in (4) is maximized when Y is also white: i.e. the spatial power spectrum at the output of the network is flat [16]. In the spatial domain, this means that all the components of Y should be uncorrelated from each other, and leads to the suggestion that the visual system may use a form of predictive coding for efficient information transmission [17] As several authors have noted [6, 18, 19] anti Hebbian ....

C. E. Shannon. Communication in the presence of noise. Proceedings of the IRE, 37:10--21, 1949.

....( 1) Z 2, to where sinc(t) si4t) for t 0 and sinc(O) 1. Remark not only that (11[ 2) is not 1 )2 but also that it does not cancel in the vicinity of its border null out of d , A (for instnco = Some other advantages, of having , 0 for ( l) T 1, from the next theorem [46] 1 2, follow Theorem 1.1 (Shannon Whittaker theorem) Let w L2( TN)2) such that O for (k,1) N N 1 , then T 1, w(m, n) sinc(x m)sinc(y for (x,y) TN) 2 and with , i fraNZ, Remark that the index d, of sincg, refers to the fact that, when the Fourier transform is ....

....the sinc interpolation (or zero padding ) which consists in filling Most of the results stated in this part are covered in [31] 35 CHAPTER 3. INTRODUCTION the lost part of the spectral domain with 0. This latest is often considered as optimal because of Shannon sampling Theorem (see [46] or Section 1.1. The shortcoming of this method is that the induced filter oscillates and is poorly localized in space domain (which yields a large algorithmic complexity for a linear interpolation) Some have reduced the algorithmic complexity (see [50] in order to compute it, but most of the ....

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C. E. Shannon. Communication in the presence of noise. Proceedings of the IRE, 37:10 21, 1949.

....[1, 6, 7] for piecewise polynomial signals which take into account the extra degrees of freedom due to the bandlimitedness. 1. INTRODUCTION Sampling of bandlimited signals has been a subject of interest to the sampling community for more than half a century [4] The well known sampling theorem [2] states that a continuous time signal x(t) bandlimited to [ #m , #m ] is uniquely represented by a uniform set of samples x[n] x(nT ) taken T seconds apart, if the sampling rate is greater or equal to the bandwidth of the signal, that is, 2# T # 2#m . But not all signals are bandlimited. ....

C. E. Shannon. Communications in the presence of noise. Proc. of the IRE, 37:10--21, January 1949.

....formulation of derivative sampling, landmark warping, and tomographic reconstruction. 111 6.2 Introduction 6.2.1 Perfect reconstruction Reconstructing a signal from its samples is one of the most fundamental tasks in signal processing. The classical sampling theorem presented by Shannon [106] states that a bandlimited function f in (whose frequency spectrum is limited by the Nyquist frequency #max = # T ) can be perfectly reconstructed from its regularly spaced (ideal) samples s j = f in (jT ) Reconstruction is carried out by convolving the samples with a sinc kernel: f out (x) f ....

....to improve the readability of this Chapter, we have chosen to present here mostly the results, in a directly applicable form. The general description of the 115 Table 6.1: Various reconstruction methods and some of their generic features. See discussion in Section 6.2. 6 Shannon [106] Papoulis [107] Wavelets, splines RBF [72] This paper Sampling type ideal generalized generalized ideal generalized Sampling locations uniform uniform uniform non uniform non uniform Reconstruction space bandlimited bandlimited splines wavelets implicit implicit ....

C.E. Shannon, "Communication in the presence of noise," in Proc. IRE, Jan. 1949, vol. 37, pp. 10--21.

....In other words, interpolation is convolution of the sampled data with some filter. Provided that the sampled function is bandlimited (i.e. it only contains frequencies different from zero within a certain interval m ; m ) and that it was sampled properly (according to Shannons sampling theorem [Shann49]) the function can be reconstructed exactly from its samples with the filter [Oppen75] sinc x sin x x if x 0 1 if x 0 (12) This filter has infinite spatial extend, so some finite approximation has to be used in practice, which will not, however, reconstruct the original function exactly. A ....

Claude E. Shannon. Communication in the presence of noise. Proceedings of the IRE, 37:10--21, January 1949.

....[1, 6, 7] for piecewise polynomial signals which take into account the extra degrees of freedom due to the bandlimitedness. 1. INTRODUCTION Sampling of bandlimited signals has been a subject of interest to the sampling community for more than half a century [4] The well known sampling theorem [2] states that a continuous time signal x(t) bandlimited to [ Gamma m ; m ] is uniquely represented by a uniform set of samples x[n] x(nT ) taken T seconds apart, if the sampling rate is greater or equal to the bandwidth of the signal, that is, 2=T 2 m . But not all signals are bandlimited. ....

C. E. Shannon. Communications in the presence of noise. Proc. of the IRE, 37:10--21, January 1949.

...., not just (his equations (17) and (18) That result, and its counterpart for the capacity of a power constrained channel with additive colored Gaussian noise, have come to be known as the water pouring formulas of information theory. In this generality the channel formula is attributable to [12] and the source formula to Pinsker [13] 14] We shall call them the Shannon Kolmogorov Pinsker (SKP) water pouring formulas. They generalize the formulas given by Shannon in 1948 for the important case in which the spectrum of the source or of the channel noise is flat across a band and zero ....

....is flat across a band and zero elsewhere. The water pouring formulas were rediscovered independently by several investigators throughout the 1950 s and 1960 s. B. The Water Table Here is a simple way of obtaining the SKP water pouring formula for the MSE information rate of a Gaussian source [12]. The spectral representation theorem lets us write any zero mean stationary random process for which in the form where is a random process with zero mean, uncorrelated increments. Hence, if and are two disjoint sets of frequencies, the zero mean random processes and defined by and satisfy ....

C. E. Shannon, "Communication in the presence of noise," Proc. IRE, vol. 37, pp. 10--21, 1949.

....of PSK (phase shift keying) and the maximum minimum distance property of the simplex configuration. Also, he gives a geometric interpretation of the threshold effect in frequency and pulse modulation systems, which is the same as the famous explanation in Shannon s 1949 Gaussian channel paper [104]. We see from (3) that the essential data processing step is the formation of the integral , which can be implemented either by correlating the stored signal against the received signal , or as is easily checked, passing through a filter matched to (i.e. one with impulse response ) and then ....

C. E. Shannon, "Communication in the presence of noise," Proc. IRE, vol. 37, pp. 10--21, 1949.

....to which a large class of high dimensional probability density functions can be well approximated with uniform densities [13, pp. 73, 285] 14] For example, data drawn from an uncorrelated Gaussian density tend to be uniformly distributed in a thin spherical shell, if the dimension is high [15], whereas the multidimensional Laplacian density can be approximated by a uniform density on the surface of a pyramid (hyperoctahedron) 16] The tendency towards uniform distributions has been successfully employed in several applications. Competitive lattice quantizers have been designed for ....

C. E. Shannon, "Communication in the presence of noise," Proceedings of the I.R.E., vol. 37, no. 1, pp. 10--21, Jan. 1949.

....30 15 0.938 BCH [15,7,5] 0.467 10 9 0 108 18 0.844 BCH [31,6,15] 0.194 30 26 0 62 31 0.969 BCH [31,11,11] 0.355 22 21 0 2,046 186 0.999 BCH [31,16,7] 0.516 14 16 0.56 42,284 155 0.645 BCH [31,21,5] 0.677 10 11 6 1 10 2 . 107,198 186 0.051 BCH [63,7,31] 0.111 62 57 0 126 63 0. 984 BCH [63,10,27] 0.159 54 54 0 1,022 196 0.998 BCH [63,16,23] 0.254 46 48 0 65,534 1,890 1.000 BCH [63,18,21] 0.286 42 46 5 5 10 3 . 262,139 1,452 1.000 BCH [63,24,15] 0.381 30 40 0.68 15,840,940 651 0.944 BCH [63,30,13] 0.476 26 34 0.75 695,053,516 1,764 0.647 BCH [63,36,11] 0.571 22 28 0.22 ....

....pp. 198 203] That the method can produce quite strong results is to some extent explained by Pierce s results [25] according to which the Gilbert Varshamov bound 7 is tight for almost all binary linear block codes, if n C ( is large. Hence, a random code is a good code. See also [23] and [27] regarding the error probability of random codes. We will now verify the observation in Section V, that G C ( is close to 1 for R C ( 1 2 and large n C ( for random codes. We first give a theorem about a random codeword in a random code in L n R , which denotes the set of all ....

C. E. Shannon, "Communication in the presence of noise," Proceedings of the I.R.E., vol. 37, no. 1, pp. 10--21, Jan. 1949.

.... Often attributed to Whittaker, Koteln ikov, and Shannon, a more precise statement of this so called WKS sampling theorem is that a real low pass signal, whose Fourier transform is limited to the range , can be recovered from its samples taken uniformly at the rate (the Nyquist rate) or higher [1]. Sampling a signal uniformly at causes the resulting spectrum to contain multiple copies of the original spectrum located with uniform spacing of between adjacent copies. Hence the choice guarantees no overlaps in the sampled spectrum, and thus allows recovery of the original signal by a ....

C. E. Shannon, "Communication in the presence of noise," Proc. IRE , vol. 37, pp. 10--21, Jan. 1949.

.... of subchannels described by Figure 2 is the mutual information I defined as I = N X i=1 log 2 1 a i 2 E[x i 2 ] oe 2 : 1) The value of I above is less than the channel capacity achieved by choosing the values of fE[x i 2 ]g to have the optimal water pouring distribution [4]. Since the subcarrier SNRs are not known at a broadcast transmitter, the optimal distribution of power is not possible. Furthermore, each channel over which data is being broadcast has a different optimal power distribution. Shannon s fundamental coding theorem ensures that for each AWGN ISI ....

C. E. Shannon. Communication in the Presence of Noise. Proceedings of the IRE, 37:10--21, January 1949.

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Shannon, C.: Communication in the presence of noise. Proc. Inst. of Radio Engineers 37 (1949) 10--21

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C. Shannon, "Communications in the presence of noise," in Proc.of the IRE, vol. 37, pp. 10--21, 1949.

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C. E. Shannon, "Communications in the presence of noise," Proc. IRE, vol. 37, pp. 10--21, Jan. 1949.

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C.E. Shannon, "Communications in the presence of noise," Proceedings of the IRE, Vol. 37, pp. 10-21, January 1949. 17

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C. E. Shannon, "Communications in the presence of noise", Proceedings of the IRE, Vol. 37, pp. 10-21, January 1949.

No context found.

Shannon, C.: Communication in the presence of noise. Proc. Inst. of Radio Engineers 37 (1949) 10--21

No context found.

C. E. Shannon, "Communication in the presence of noise.," Proc. Institute of Radio Engineers, vol. 37, pp. 10--21, 1949.

No context found.

C.E. Shannon. Communication in the Presence of Noise. Proceedings of the IRE, 37:10--21, 1949.

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Shannon, C.E., Communications in the Presence of Noise, Proceedings of the IRE, Vol. 37, pp. 10-21, January 1949. 30

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C. E. Shannon, "Communication in the presence of noise," Proc. IRE, vol. 37, pp. 10--21, 1949.

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C. E. Shannon, "Communication in the presence of noise," in Proc. of the IRE, 37, pp. 10--21, 1949.

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Shannon, C.: Communication in the presence of noise. Proceeding s of the IRE 37 (1949) 10--21 [reprinted in: Proceedin g of the IEEE 86 (1998) 447--457].

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C.E. Shannon, Communication in the presence of noise. Proceedings of the IRE 37(1) (1949) 10-21.

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C.E. Shannon, Communication in the presence of noise, Proc. IRE 37(1949), pp. 10-21.

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Shannon, C.E., Communications in the presence of noise, Proceedings of the IEEE, 37, 10-21 (1949)

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