Abstract:
Methods are disclosed for creation of vector signaling codes having reduced or minimized electromagnetic emissions when transmitted over multiple conductor communications media such as microstrip channels. Particular code formulation constraints are shown to correlate with physical propagation characteristics leading to beneficial cancellation of far fields from adjacent signal conductors, and the number of such constraints may be chosen to balance EMI characteristics with information encoding capacity in the resulting code.

Description:
This application claims priority to U.S. application Ser. No. 61/934,800 filed Feb. 2, 2014 titled “Low EMI Signaling for Parallel Conductor Interfaces”, the contents of which are incorporated herein by reference. 
    
    
     REFERENCES 
     The following references are herein incorporated by reference in their entirety for all purposes: 
     U.S. Patent Publication No. 2011/0268225 of U.S. patent application Ser. No. 12/784,414, filed May 20, 2010, naming Harm Cronie and Amin Shokrollahi, entitled “Orthogonal Differential Vector Signaling” (hereinafter “Cronie I”); 
     U.S. Patent Publication No. 2011/0302478 of U.S. patent application Ser. No. 12/982,777, filed Dec. 30, 2010, naming Harm Cronie and Amin Shokrollahi, entitled “Power and Pin Efficient Chip-to-Chip Communications with Common-Mode Resilience and SSO Resilience” (hereinafter “Cronie II”); 
     U.S. Provisional Patent Application No. 61/812,667, filed Apr. 16, 2013, naming John Fox, Brian Holden, Ali Hormati, Peter Hunt, John D Keay, Amin Shokrollahi, Anant Singh, Andrew Kevin John Stewart, and Giuseppe Surace, entitled “Methods and Systems for High Bandwidth Communications Interface” (hereinafter called “Fox I”); The following reference is also cited in this document: 
     Paul R. Clayton, Introduction to Electromagnetic Compatibility, Wiley-Interscience, 2006, herein referred to as [Clayton 2006]. 
     BACKGROUND 
     Communications interfaces using multiple wire connections are well represented in the art. Historically, parallel wire cables were supplanted by twisted pair and shielded cables at low frequencies and stripline and microstrip transmission lines at higher frequencies, in an effort to minimize electro-magnetic interference (EMI) emission. Vector signaling is a method of signaling. With vector signaling, pluralities of signals on a plurality of wires are considered collectively, although each of the plurality of signals may be independent. One encoded unit of the vector signaling code is termed a “codeword”, and each of the collective signals of a codeword is referred to as a component, with the number of signals communicated on the plurality of wires representing one codeword referred to as the “dimension” of the vector. 
     With binary vector signaling, each component takes on a coordinate value (or “coordinate”, for short) that is one of two possible values. As an example, eight single ended signaling wires may be considered collectively, with each component/wire taking on one of two values each signal period. A “code word” of this binary vector signaling is one of the possible states of that collective set of components/wires. A “vector signaling code” or “vector signaling vector set” is the collection of valid possible code words for a given vector signaling encoding scheme. A “binary vector signaling code” refers to a mapping and/or set of rules to map information bits to binary vectors. 
     With non-binary vector signaling, each component has a coordinate value that is a selection from a set of more than two possible values. A “non-binary vector signaling code” refers to a mapping and/or set of rules to map information bits to non-binary vectors. The ability of a vector signaling code to encode data is constrained by the number of codewords in the code, with larger codeword sets being capable of encoding more data. The ratio of encoded data capacity (i.e., the binary logarithm of the number of codewords) to the dimension of the codewords (or equivalently, to the number of actual bits transmitted via the codeword versus the number of physical pins or wires needed to communicate that number of bits in parallel,) is known as the pin-efficiency of the vector signaling code. Examples of vector signaling methods are described in Cronie I, Cronie II, and Fox I. 
     Viewed as a layered networking model, significant effort has been expended at the physical layer to optimize wiring, transmission line characteristics, signal amplitudes, etc. to minimize undesirable signal emission. An analysis of signal coding methods indicates that significant EMI reduction may be obtained by careful selection of appropriate signal encodings, such as particular vector signaling codes, for signals transmitted over such physical media. 
     BRIEF DESCRIPTION 
     Properties and the construction method of signaling schemes are disclosed for codes that reduce electromagnetic interference produced by several parallel conductors when excited with that particular signaling. Such codes makes easier to fulfill the EMI/EMC requirements set by international regulations. 
     First, the E-field produced by several equally spaced parallel conductors excited with some signaling scheme is derived. Second, the construction of codes is disclosed for a signaling scheme to minimize electromagnetic interference (EMI) produced by the currents in the conductors, and specific examples of code construction are provided. Finally, electromagnetic simulations of the E-field are documented which support the theoretical findings and claims. 
    
    
     
       BRIEF DESCRIPTION OF FIGURES 
         FIG. 1  illustrates the calculation of the far fields for two wire currents. 
         FIG. 2  is an illustration of a four conductor stripline transmission line. 
         FIG. 3  is a detail of the four conductor stripline transmission line of  FIG. 2 . 
         FIG. 4  is a graph showing calculated emissions for different codewords transmitted over the transmission line of  FIGS. 2 and 3   
         FIG. 5  is a graph comparing EMI emissions for Differential NRZ signaling and ENRZ signaling over the same wires at equivalent delivered data rates. 
         FIG. 6  depicts a flowchart of a process in accordance with at least one embodiment. 
     
    
    
     DETAILED DESCRIPTION 
     We consider a system with a number N of equally spaced parallel conductors, each carrying some current I l ,l=0 . . . N−1. Each conductor can be modeled as a Hertzian dipole, and the E-field (far field) generated by the current Î l  of the l th  conductor at distance r l  being expressed according to [Clayton 2006] as: 
                       E   ^     θ     =       M   ^     ⁢       I   ^     l     ⁢       ⅇ       -     jβ   0       ⁢     r   l           r   l       ⁢     F   ⁡     (   θ   )                 [     Eqn   .           ⁢   1     ]               
where Î l  is the current at the center of the conductor and the factor F(θ) has a maximum value of unity and presents the θ variation of the antenna (E-field) pattern. β 0 =2π/λ 0 =2πf/c where f is the frequency in Hz and c=3·10 8  is the speed of light in vacuum. We herein denote θ and φ as identifying the two angles of a spherical coordinate system, in which φ is the angle with respect to the axis on which the conductor is aligned, and θ is the angle around the axis of the conductor.  FIG. 1  from [Clayton 2006] illustrates an example of a system with 2 parallel conductors using this notation.
 
     One should note that, by symmetry, the pattern of the antenna is independent of φ but the pattern of a pair of antennas (conductors) can be a function of φ. The term {circumflex over (M)} is a function of the antenna type. For a Hertzian dipole, the terms {circumflex over (M)} and F(θ) are: 
                           M   ^     =     j   ⁢           ⁢   2   ⁢           ⁢     π   ⨯     10     -   7         ⁢   f   ⁢           ⁢   L                   F   ⁡     (   θ   )       =     sin   ⁢           ⁢   θ                   [     Eqn   .           ⁢   2     ]               
where L represents the length of the conductors. The above formulas are valid only for current segments that are very short electrically, which allows us to assume that the current at all points along the antenna is the same (magnitude and phase). For these linear, wire-type antenna, the radiated electrical field achieves its maximum amplitude broadside to the antenna, that is θ=90°. Hence, the radiated electric field Ê θ  is parallel to the conductor axis, and F(θ)=1.
 
     The total E-field generated by the N parallel conductors placed equidistantly at distance s from each other will be the superposition of the fields generated by each conductor in separation: 
                       E   ^     θ     =         ∑     l   =   0       N   -   1       ⁢       E   ^       θ   ,   l         =       M   ^     ⁢       ∑     l   =   0       N   -   1       ⁢         I   ^     l     ⁢       ⅇ       -     jβ   0       ⁢     r   l           r   l                       [     Eqn   .           ⁢   3     ]               
In terms of the distance r from the midpoint between the conductors to the measurement point, the N rays can be written as:
 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             r 
                             l 
                           
                           = 
                           
                             r 
                             - 
                             
                               
                                 ( 
                                 
                                   
                                     
                                       N 
                                       - 
                                       1 
                                     
                                     2 
                                   
                                   - 
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                                 ) 
                               
                               ⁢ 
                               s 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
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                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               ϕ 
                             
                           
                         
                       
                       
                         
                           l 
                           = 
                           
                             0 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             … 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             N 
                           
                         
                       
                     
                   
                   - 
                   1 
                 
               
               
                 
                   [ 
                   
                     Eqn 
                     . 
                     
                         
                     
                     ⁢ 
                     4 
                   
                   ] 
                 
               
             
           
         
       
     
     Since we are considering only the far field, the radial distances from the N conductors are approximately equal, that is, r 0 ≅r 1 ≅ . . . ≅r N−1 ≅r. We may substitute r 0 =r 1 = . . . =r N−1 =r into the denominator of the E-field formula, but we should not make the substitution into the e −jβ     0     r     l    terms for the following reason. This term can be written as e −j2πr     l     /λ     0    , and its value depends not on the physical distance r l  but on the electrical distance r l /λ 0 . Therefore, even though two rays r 0  and r 1  may be approximately equal, the exponential term can depend significantly on the difference in electrical distances. For example, if f=3GHz (λ 0 =10 cm) and r 0 =10 m, r 1 =10.5 m, the quantities β 0 r 0  and β 0 r 1  differ by 180° , so the fields are of opposite phases! For our system, the maximum of the E-field will occur in the plane of the wires and on a line perpendicular to the wires (φ=0°, 180°) . With the new substitutions, we obtain: 
     
       
         
           
             
               
                 
                   
                     
                       E 
                       ^ 
                     
                     max 
                   
                   = 
                   
                     
                       M 
                       ^ 
                     
                     ⁢ 
                     
                       
                         ⅇ 
                         
                           
                             - 
                             
                               jβ 
                               0 
                             
                           
                           ⁢ 
                           r 
                         
                       
                       r 
                     
                     ⁢ 
                     
                       
                         ∑ 
                         
                           l 
                           = 
                           0 
                         
                         
                           N 
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                         
                           
                             I 
                             ^ 
                           
                           l 
                         
                         ⁢ 
                         
                           ⅇ 
                           
                             
                               j 
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     
                                       N 
                                       - 
                                       1 
                                     
                                     2 
                                   
                                   - 
                                   l 
                                 
                                 ) 
                               
                             
                             ⁢ 
                             
                               β 
                               0 
                             
                             ⁢ 
                             s 
                           
                         
                       
                     
                   
                 
               
               
                 
                   [ 
                   
                     Eqn 
                     . 
                     
                         
                     
                     ⁢ 
                     5 
                   
                   ] 
                 
               
             
           
         
       
     
     We assume that the currents Î l  are produced by a driver which generates on each of the conductors l=0 . . . N−1 a current level proportional with the transmitted codeword c l  on that particular wire, i.e., Î l =α*c l , where α is some real valued scalar constant which depends on the transmitter driver architecture. We can also rewrite the quantity in the exponent as: 
                       j   ⁡     (         N   -   1     2     -   l     )       ⁢     β   0     ⁢   s     =         j   ⁡     (         N   -   1     2     -   l     )       ⁢       2   ⁢   π     λ     ⁢   s     =       j   ⁡     (         N   -   1     2     -   l     )       ⁢       2   ⁢   π   ⁢           ⁢   f     c     ⁢   s               [     Eqn   .           ⁢   6     ]               
In the following we use the Taylor series expansion to rewrite the exponential term:
 
                     e   x     =         ∑     n   =   0     ∞     ⁢       x   n       n   !         =     1   +   x   +       x   2       2   !       +       x   3       3   !       +       x   4       4   !       +   …               [     Eqn   .           ⁢   7     ]               
For conductor inter-spacing in the order of a few hundred microns, and frequencies in the range of a few GHz, the quantity in the exponent is much smaller than unity. For example, for s=100 μm, f=1 GHz, N=4 and l=0, the absolute value of the exponent is 1.5 ·2π·10 9 ·10 −4 /(3·10 8 )≅3.14·10 −3 &lt;&lt;1. In this case, it is reasonable to assume that the first two terms from the Taylor expansion are dominant and that the contribution from the other terms to the total sum is negligible:
 
                     ⅇ       j   ⁡     (         N   -   1     2     -   l     )       ⁢     β   0     ⁢   s       =     1   +       j   ⁡     (         N   -   1     2     -   l     )       ⁢     β   0     ⁢   s     +       f   ɛ     ⁡     (       β   0     ,   s   ,   l     )                 [     Eqn   .           ⁢   8     ]               
where f ε (β 0 , s, l) is a small quantity accumulating the summation of the remaining Taylor expansion terms. Then, the magnitude of the E-field can be written as
 
                            E   ^     max          =       ⁢     α   ⁢            M   ^          r     ⁢            ∑     l   =   0       N   -   1       ⁢       c   l     ⁢     ⅇ       j   ⁡     (         N   -   1     2     -   l     )       ⁢     β   0     ⁢   s                                     ⁢     [     Eqn   .           ⁢   9     ]                 =       ⁢     α   ⁢       2   ⁢           ⁢   π   ×     10     -   7       ⁢     f   L       r     ⁢            ∑     l   =   0       N   -   1       ⁢       c   l     (     1   +       j   ⁡     (         N   -   1     2     -   l     )       ⁢     β   0     ⁢   s     +                         ⁢     [     Eqn   .           ⁢   10     ]                       ⁢       f   ɛ     ⁡     (       β   0     ,   s   ,   l     )       )                            =       ⁢     α   ⁢       2   ⁢           ⁢   π   ×     10     -   7       ⁢     f   L       r     ⁢              ∑     l   =   0       N   -   1       ⁢     c   l       +     j   ⁢           ⁢     β   0     ⁢   s   ⁢       N   -   1     2     ⁢       ∑     l   =   0       N   -   1       ⁢     c   l         -                     ⁢     [     Eqn   .           ⁢   11     ]                     ⁢       j   ⁢           ⁢     β   0     ⁢   s   ⁢       ∑     l   =   0       N   -   1       ⁢     lc   l         +       ∑     l   =   0       N   -   1       ⁢       c   l     ⁢       f   ɛ     ⁡     (       β   0     ,   s   ,   l     )                                    
If we have a codebook with P codewords, C={c 1  . . . c P }  ⊂  R N , the average E-field over all the codewords (that is, all of the conductor currents utilized in practice) will be
 
                            E   ^     max          =     2   ⁢     π   ⨯     10     -   7         ⁢       f   ⁢           ⁢   L     r     ⁢   α   ⁢     1   P     ⁢       ∑     i   =   1     P     ⁢              ∑     l   =   0       N   -   1       ⁢     c     i   ⁢           ⁢   l         +     j   ⁢           ⁢     β   0     ⁢   s   ⁢       N   -   1     2     ⁢       ∑     l   =   0       N   -   1       ⁢     c     i   ⁢           ⁢   l           -     j   ⁢           ⁢     β   0     ⁢   s   ⁢       ∑     l   =   0       N   -   1       ⁢     l   ⁢           ⁢     c     i   ⁢           ⁢   l             +       ∑     l   =   0       N   -   1       ⁢       c     i   ⁢           ⁢   l       ⁢       f   ɛ     ⁡     (       β   0     ,   s   ,   l     )                              [     Eqn   .           ⁢   12     ]               
where c il  represents the i th  codeword component on wire l. Note that the term fε(β 0 , s, l) leads to summations of the form Σ l=0   N−1  c il , Σ l=0   N−1  lc il , Σ l=0   N−1  l 2 c il , etc.
 
Low EMI Code Properties and Design
 
     The expression of the E-field derived in Eqn. 12 shows that in order to design low emissions (EMI) codes one needs to construct them such that to minimize the summations Σ l=0   N−1  c il , Σ l=0   N−1  lc il , Σ l=0   N−1  l 2 c il , etc. The first two terms are more critical to this minimization, as they are the dominant quantities in the E-field expression. As commonly used in descrete mathematics, in the following we will associate with a codeword [c 0 , c 1 , . . . , c N−1 ] a polynomial of degree N−1, p(x)=c N−1 x N−1 + . . . +c 1 x+c 0 . 
     The codes which minimize the EMI can be designed as follows. As a first criterion, to satisfy the condition Σ l=0   N−1  c il =0 (i.e. first order equal to zero,) it follows directly that the code polynomial should evaluate to 0 at 1, i.e., that the code polynomial should be divisible by (x−1). This means that we may generate a first-order low EMI code, by taking any code C′ ⊂               N−2 , interpreting its codewords as polynomials of maximum degree N−2, and multiplying this polynomial by (x−1). The coefficients of the resulting polynomial will give us the desired new codewords.
     As a second criterion, to satisfy the conditions Σ l=0   N−1  c il =0 and Σ l=0   N−1  lc il =0 (first order and second order equal to zero,) the codeword polynomials should be divisible by(x− 1 ) 2 . This follows from noting that the previous two conditions are equivalent to Σ l=0   N−1  c il =0 and Σ l=0   N−1  (l−1)c il =0. 
     In general, if we want to design a code which satisfies Σ l=0   N−1  l k c il =0, ∀ k=0 . . . M−1, M≧1, we can construct it by taking any code C′ ⊂               N−M−1  defined by a polynomial of maximum degree N−M−1 and then multiply the codeword polynomials by (x−1) M . The coefficients of the resulting polynomial will give us the desired new codewords.
     A variation of this construction strategy constrains one or more lower-order terms to be zero, and then bounds the next term to be less than a fixed amount. 
     This construction mechanism permits the creation of a code having the desired EMI characteristics from an existing code. In the general case, the new code will have a large dimension (i.e. require one or more signal conductors) than the original code from which it was created. In other words, the new code will in general have a lower pin-efficiency than the original code, but will result in much lower EMI emissions. 
     Low EMI Code Examples 
     As a first example of how the described method leads to reduced EMI codes, consider the case of two parallel conductors, N=2, M=1. We consider C′={±1} and the generator polynomial (x−1). The low EMI code is defined by ±1(x−1), and its corresponding codewords will be [1, −1] and [−1,1] which is in fact differential signaling, which is known in the art to be a low-emission encoding for wire pairs. This code satisfies Σ l=0   N−1  c il =0. 
     In a second example consider three conductors, N=3, M=2. We consider C′={±½} The low EMI code is defined by ±½(x−1) 2 =±½(x 2 −2x+1) and its corresponding codewords are [½, −1, ½] and [−½, 1, −½]. Note that this code satisfies both Σ l=0   N−1  c il =0 and Σ l=0   N−1  lc il =0. 
     In a third example of the described method for four conductors, N=4, M=2. We consider C′={±1}. The low EMI code is chosen to be defined by ±1(x+1)(x−1) 2 =±(x 3 −x 2 −x+1) and its corresponding codewords are [1, −1, −1, 1] and [−1, 1, 1, −1]. Note that this code satisfies both Σ l=0   N−1  c il =0 and Σ l=0   N−1  lc il =0 and differs from the conventional practice of using two parallel differential pairs; conventional practice would suggest use of the codewords [1, −1, 1, −1], [−1, 1, −1, 1], [1, −1, −1, 1], and [−1, 1, 1, −1] (that is, with the two differential pairs operating independently,) whereas the described method identifies only two of the four differential pair codewords as having low EMI characteristics. 
     In one embodiment a method  600  is described with respect to  FIG. 6 . At block  602  a set of data bits representing information is received. At block  604  the data bits are mapped to low EMI codewords in a set of low EMI codewords, wherein the set of low EMI codewords when represented as a polynomial are divisible by the polynomial (X−1) M  where M is at least 1. At block  606  the set of low EMI codewords are transmitted on a multiwire system. The method may generate the set of low EMI codewords by modulating an initial set of codewords represented as a polynomial of maxiumum degree N−M− 1  and a polynomial (X−1) M  where N is a number of wires in the multiwire system and M is an order of electromagnetic interference reduction. The method may utilize a system wherein M is at least 2, representing a plurality of orders of electromagnetic interference reduction. The method may utilize a system wherein the initial set of codewords are formed using a Hadamard matrix with a size of at least 3. The method may utilize a system wherein the Hadamard matrix is an ENRZ code. The method may utilize a system wherein N is at least 3. 
     Constructing Low EMI Codes 
     One embodiment of a constructed Low-EMI code begins with an initial three symbol code comprised of all possible combinations of the symbol alphabet {+1, −1}, in other words a three-bit binary code. With three values each having two possibilities, this code has eight codewords. As taught by [Cronie I] and [Cronie II], the Hadamard transform of size 4 (also known as the H4 matrix) may be used to transform this initial three bit code into a new code having eight codewords representing all permutations of [+3, −1, −1, −1] and [−3, +1, +1, +1]. Also known as Ensemble NRZ or ENRZ, the H4 codewords are often shown with a constant ⅓ scaling factor, producing a symbol alphabet of {+1 +⅓, −⅓, −1}. The H4 or ENRZ code is balanced, thus satisfying Σ l=0   N−1  c il =0. 
     It should be noted that transmission of the initial three bit binary code requires three wires, and its detection requires three receivers each comparing one wire against a fixed reference. Thus, the initial code has no inherent protection against common mode noise. In comparison, the transformed code requires one additional wire but, as taught by [Fox I], may be detected by three comparators performing only differential comparisons among the wires, thus providing common mode noise rejection. 
     A second embodiment uses a Hadamard matrix of size 8 (also known as H8 matrix) as taught by [Cronie I] and [Cronie II] multiplied by the vectors (0, 0, 0, ±1, 0, ±1, ±1, ±1) to produce a code capable of transmitting 4 bits on 8 wires, with both Σ l=0   N−1  c il =0 and Σ l=0   N−1  lc il =0, that is, both first order and second order equal to zero. Multiplying H8 by [0, 0, . . . , 0, ±1] produces a code having first, second, and third order 0, but with only two codewords, thus being capable of encoding only one bit on 8 wires. 
     A further embodiment uses the Hadamard matrix of size sixteen (H16) in a similar fashion; Multiplying H16 by the vector (0, ±1, ±1, . . . , ±1) gives a first order 0 and can encode 15 bits on 16 wires. Multiplying H16 by the vector [0, 0, 0, ±1, 0, ±1, ±1, ±1, 0, ±1, ±1, ±1, ±1, ±1, ±1, ±1] gives a first and a second order 0, and can encode 11 bits on 16 wires. Multiplying H16 by [0, 0, 0, 0, 0, 0, 0, ±1, 0, ±1, 0, 0, 0, ±1, 0, ±1, ±1, ±1] gives a code having first, second, and third order 0, and capable of encoding 6 bits on 16 wires. Finally, multiplying H16 by [0, 0, . . . , 0, ±1] produces a code having first, second, third, and fourth order 0, but with only two codewords, thus being capable of encoding only one bit on 16 wires. 
     Thus, the described method may be used to derive new codes having Low-EMI characteristics from an initial code, with a wide range of trade offs available between EMI characteristics and pin-efficiency for the resulting code. 
     Electromagnetic Simulation Results 
     In order to confirm these theoretical results, the 3D electromagnetic simulation software CST Microwave Studio (CST MWS) was used to evaluate the E-field (farfield) produced by a stripline structure when excited with several signaling schemes and confirm their emissions follow the tendencies derived earlier. Due to their low insertion losses and good impedance control and matching, striplines with vias represent a preferred structure for transmitting very high speed electrical signals. We choose N=4 and the considered stripline transmission lines are depicted in  FIGS. 2 and 3 . For some chosen codewords, in Table 1 we compute the quantities Σ l=0   N−1  c il  and Σ l=0   N−1  lc il . 
     
       
         
               
             
               
               
               
               
             
           
               
                 TABLE 1 
               
             
             
               
                   
               
               
                 E-field for various codewords 
               
             
          
           
               
                   
                 Codeword 
                 | Σ l=0   N−1  C il  | 
                 | Σ l=0   N−1  lC il  | 
               
               
                   
                   
               
               
                   
                 [1, −1, −1, 1] 
                 0 
                 0 
               
               
                   
                 [⅓, −1, ⅓, ⅓] 
                 0  
                 ⅔ 
               
               
                   
                 [0, −1, 1, 0] 
                 0 
                 1 
               
               
                   
                 [1, −⅓, −⅓, −⅓] 
                 0 
                 2 
               
               
                   
                 [1, 0, −1, 0] 
                 0 
                 2 
               
               
                   
                 [1, −1, 1, −1]  
                 0 
                 2 
               
               
                   
                 [1, 0, 0, −1] 
                 0 
                 3 
               
               
                   
                 [0, 1, 1, 0] 
                 2 
                 3 
               
               
                   
                 [1, 1, 1, 1] 
                 4 
                 6 
               
               
                   
                   
               
             
          
         
       
     
     In  FIG. 4  we plot the simulated radiated emissions curves for each of the codewords given in Table 1. We notice that in average they respect the ordering given by our previous theoretical calculations. More specifically, the codeword [1, −1, −1, 1] has the lowest emissions across most of the plotted frequencies range, followed by the [⅓, −1, ⅓, ⅓]and [0, −1, 1, 0] codewords. However, we need to stress at this point the fact that there are at least two sources of imperfections with respect to the theoretical model which can cause and explain the deviations of the simulated results from the theoretical findings. First of all, the structure itself it is not truly an exact representation of the parallel conductors model used when deriving our theoretical results. Secondly, the neglected terms from the Taylor series expansion might sum constructively or destructively depending on the chosen codewords, changing the value of the calculated E-field. 
       FIG. 5  is a graph comparing EMI emissions for Differential NRZ signaling and ENRZ signaling over the same wires at equivalent delivered data rates. The clock frequency for the Differential NRZ signaling is 28 GHz (translating to 112 Gbps delivered data rate over all 4 wires), and the clock frequency for ENRZ is 18.66 GHz (also translating to 112 Gbps delivered data rate over all 4 wires). As may be seen, although both signaling methods are below the FCC emissions limit in this simulation, ENRZ produces significantly less emission than Differential NRZ, especially in the regions between 17-22 GHz and above 30 GHz.