How to Build a Logical Shifter in Hardware

Shifters

Digital Building Blocks

Sarah L. Harris , David Harris , in Digital Design and Computer Architecture, 2022

five.2.five Shifters and Rotators

Shifters and rotators move bits and multiply or divide by powers of 2. As the proper name implies, a shifter shifts a binary number left or right by a specified number of positions. Several kinds of usually used shifters exist:

Logical shifter—shifts the number to the left or correct and fills empty spots with 0's.

Example: 11001 >> 2 = 00110; 11001 << 2 = 00100

Arithmetic shifter—is the same every bit a logical shifter, but on right shifts fills the nearly significant bits with a copy of the old well-nigh significant bit (msb). This is useful for multiplying and dividing signed numbers (encounter Sections five.2.half dozen and five.two.7). Arithmetic shift left is the aforementioned as logical shift left.

Example: 11001 >>> 2 = 11110; 11001 << 2 = 00100. The operators <<, >>, and >>> typically indicate shift left, logical shift correct, and arithmetic shift right, respectively.

Rotator—rotates a number in a circle such that empty spots are filled with bits shifted off the other terminate.

Example: 11001 ROR two = 01110; 11001 ROL ii = 00111. ROR = rotate right; ROL = rotate left.

An N-bit shifter can be built from North N:1 multiplexers. The input is shifted by 0 to North −1 bits, depending on the value of the log_two N-fleck select lines. Figure 5.19 shows the symbol and hardware of 4-chip shifters. Depending on the value of the two-bit shift amount shamt _1:0, the output Y receives the input A shifted by 0 to 3 bits. For all shifters, when shamt _1:0 = 00, Y = A. Do 5.22 covers rotator designs.

Figure 5.19. 4-scrap shifters: (a) shift left, (b) logical shift right, (c) arithmetic shift right

A left shift is a special case of multiplication. A left shift past Northward bits multiplies the number by 2 ^N . For example, 000011_ii << four = 110000₂ is equivalent to iii₁₀ × two^four = 48_x.

An arithmetics right shift is a special case of division. An arithmetic right shift by N bits divides the number past 2 ^N . For example, 11100₂ >>> 2 = 11111₂ is equivalent to −iv₁₀/2² = −1₁₀.

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Recent Advances in Magnetic Insulators – From Spintronics to Microwave Applications

Gopalan Srinivasan , ... Igor 5. Zavislyak , in Solid State Physics, 2013

3.one Ferrite Dielectric Phase Shifters

A magnetic-field tunable W-ring phase shifter based on dielectric resonance in barium hexaferrite is discussed. A stage shift of threescore° with depression losses was achieved at depression bias magnetic field.

Tunable phase shifters are critical components in phased array radars for beam steering, besides as in wireless and satellite communication systems [20]. Stage shifters for the W-band, in particular, are of interest for apply in phased array transmitters and receivers for car radars [21]. Ferrite phase shifters based on Faraday rotation have the advantages of low-insertion loss and high power-handling capability. Ferrite phase shifters could also exploit the fast variation of magnetic permeability near FMR frequency; hence their operating frequency is divers by the FMR frequency range of a ferrite material. Spinel ferrites or garnets are not suitable for millimeter moving ridge phase shifters due to the large external magnetic field necessary to operate at loftier frequencies. Pure, Sc-doped, or Al-doped barium or strontium hexagonal ferrites with M-type structures have high uniaxial anisotropy fields and are advisable materials for FMR-based phase shifters in the Ka- to V-bands [22–25]. But devices at W-band would nevertheless require a very large bias magnetic field. Ane possible solution for phase shifters in the W-band is the use of Al-substituted M-type hexaferrites [18,23], which have much larger uniaxial magnetocrystalline anisotropy field than BaFe₁₂O₁₉ (BaM), but substitution of Al for Fe increases losses. An alternating to FMR-based devices is the utilization of dielectric resonances BaM discussed in Section ii.2. Such resonances occur at a much college frequency than FMR in gyrotropic resonators with rotational symmetry and could be tuned with an external bias field H to reach a differential phase shift.

A single crystal c-plane BaM disc with bore D  =   1.24   mm and thickness S  =   0.28   mm was chosen since rotational symmetry for the sample is a key requirement. The uniaxial anisotropy field for the ferrite was 16.8   kOe. The sample was mounted in a WR-10 waveguide flange and sandwiched between a 30-μm thick dielectric polyethylene layer and a foam slab. The dielectric layer serves two purposes; by moving the BaM deejay abroad from the metal surface ane decreases the high frequency losses and slightly increases the frequency of the main Eastward _11δ dielectric style. For H  =   0, the frequencies f of East-type dielectric modes are given by [xix]

(ten.1) $tan\left({\beta }_z\mathrm{Southward}\right)=\left(tan\mathrm{h}\left({\beta }_{1z}{d}_1\right)+tan\mathrm{h}\left({\beta }_{\mathrm{ane}z}{d}_{\mathrm{ii}}\right)\right)/\left(\frac{{\beta }_z}{{\beta }_{1z}{\varepsilon }_{\perp }}-\frac{{\beta }_{\mathrm{one}z}{\varepsilon }_{\perp }}{{\beta }_z}tan\mathrm{h}\left({\beta }_{1z}{d}_1\right)tan\mathrm{h}\left({\beta }_{1z}{d}_2\right)\right)$

Hither ${\beta }_z=\sqrt{{\left(\frac{\mathrm{two}\mathit{\pi f}}{c}\right)}^{\mathrm{ii}}{\varepsilon }_{\perp }-\frac{{\varepsilon }_{\perp }}{{\varepsilon }_{||}}{\beta }^{\mathrm{two}}}$ , ${\beta }_{1z}=\sqrt{{\beta }^{\mathrm{ii}}-{\left(\frac{\mathrm{two}\mathit{\pi f}}{c}\right)}^2}$ , ɛ _⊥ and ɛ _|| are the transverse and longitudinal dielectric permittivity respectively, c is the speed of light, d _one and d ₂ are thickness of air space between resonator and metal planes in a higher place and below, and β  =   twoA _nm /D, A _nm is an m-th root of Bessel functions ${J^{\prime }}_n\left(\mathrm{ten}\right)=0$ . Due to nonreciprocity of a magnetized ferrite medium with respect to two rotation directions, the degeneracy is removed for H  ≠   0 and their frequencies become magnetic field dependent [20, 23, 24]. When the resonator middle bespeak is placed at a quarter-width from the waveguide sidewall, the polarization of waveguide fashion across the resonator is predominantly circular (left or right, depending on direction of propagation). Hence, if straight wave at the given frequency excites the counterclockwise rotation mode, for example, the contrary moving ridge cannot excite either the counterclockwise mode (due to unfavorable polarization) or the clockwise mode (polarization is appropriate but frequency is different). Therefore such an arrangement ensures operation with merely one of the split modes and the phase shifter becomes nonreciprocal.

Calculations of nix-field Eastward _xiδ resonance frequency versus d _one using Eq. (10.i) with ɛ _⊥  = ɛ _||  =   16 for a series of thicknesses South are shown in Fig. one of [19]. It was assumed that the sample lies in a WR-ten waveguide with dimensions a  =   2.54   mm and b  =   i.27   mm then that d ₂  = b  − S  – d ₁. One could easily command aught-field frequency of East _11δ mode and hence the phase shifter operating point in the whole West-band by just varying d _i [19].

Phase shift measurements were carried out using a 75- to 110-GHz Agilent vector network analyzer. Power was applied to the sample-mounted flange that was inserted between two WR-10 waveguides. The bias magnetic field H was aligned perpendicular to the disc plane (forth the c-axis). Profiles of S ₂₁ amplitude and differential phase shift versus f, for a series of H are shown in Fig. 10.6.

Effigy x.6. Dielectric resonance absorption profiles and differential phase shift of BaM gyromagnetic resonator at different values of practical magnetic field.

Loftier-frequency fashion was chosen for the stage shift measurements since it provides larger phase shift in comparison with low-frequency mode E _{+   xiδ} . The operating frequency of eighty   GHz was chosen in order to reduce the insertion loss. Data on the differential phase shift Δφ versus H is shown in Fig. ten.7. One obtains a maximum Δφ of 60° for H  =   3200   Oe and an insertion loss of i.5–four   dB. This phase shifter is indeed nonreciprocal with Δφ upwardly to −   250° in the reverse direction while the losses are x   dB. Note that contrary to the piece of work in [3] where ferrite needs to be saturated, our resonator is based on dielectric resonance hence can operate in the unsaturated authorities.

Figure x.7. Differential phase shift and insertion loss for an lxxx-GHz BaM dielectric stage shifter.

A similar magnetically tunable passive narrow-band dissever-mode mm-wave phase shifter based on dielectric resonance in YIG was likewise investigated [nineteen]. The phase shifter frequency was in the region between two split dielectric resonances in YIG. It was shown that, under certain weather condition, the differential phase shift from the divide modes added upwards, resulting in a larger stage shift than a single-mode phase shifter. Two prototype stage shifters operating in the U- and Westward-bands at frequencies much higher than FMR in YIG were designed and characterized. Phase shifts upwards to 30° with low losses and acceptable standing wave ratio are obtained for moderate bias magnetic fields (~   one.5   kOe). Equivalent transmission-line model taking into account coupling between the split resonances was presented and at that place was reasonable agreement between theory and experiment for both insertion losses and differential phase shifts [19].

Information technology is of interest here to express the phase shift in Fig. x.seven in terms of figures of merit (FOM) such equally phase shift per unit length or phase shift per unit insertion loss. Since the dielectric resonance in the BaM disk with diameter D  =   one.24   mm and thickness S  =   0.28   mm is primarily determined by the thickness, the maximum phase shift of ~   55° corresponds to FOM   =   196° per mm or 55° per dB. For comparison, YIG phase shifter based on dielectric resonance is reported to have FOM   =   100° per mm or nineteen°per dB [19]. The FOM for BaM phase shifter is comparable to FOM for semiconductor device-based phase shifters, just information technology is inferior to MEMS stage shifters [19]. Desired phase shifts can exist accomplished by combining multiple identical disk resonators in the waveguide section. A 180° phase shift can be accomplished past placing several resonators at 3λ/4 distance between centers in a W-band waveguide.

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Silicon-Based Millimeter-wave Technology

M. Daneshmand , R.R. Mansour , in Advances in Imaging and Electron Physics, 2012

7.3 RF MEMS Phase Shifter

With RF MEMS technology rapid development, RF/microwave MEMS phase shifters take exhibited excellent RF performance, such every bit loftier isolation, high stage shifts, low insertion loss, and wide bandwidth operation features at high frequencies. They are essential components in phased-array antennas for telecommunications and radar applications ( Mousavi et al., 2008; Fakharzadeh et al., 2008). At that place are ii main designs for these components: the switched network and the distributed micro-electromechanical transmission-line approach.

The switched-network approach consists of switching different delay lines to obtain various required phase shifts (He et al., 2006). Figure 33 shows an example of a 2-b switched-line phase shifter (Tan et al., 2003). Dissimilar line lengths have been optimized to offering 0°, xc°, 180° and 270° phase shifts and continued to two University of Michigan/Rockwell Scientific SP4T switches. The switches are controlled to connect the desired path and consequently obtain the associated stage shift. The insertion loss, render loss, and the linearity of the achieved stage shifts accept been demonstrated in Figure 33(b). In designing this blazon of configuration, the switches play a disquisitional role equally they dominate the insertion loss of the entire phase shifter. For this configurations, the linearity of the phase shift, as well as isolation of the lines, are disquisitional.

Effigy 33. The 2-b University of Michigan/Rockwell Scientific SP4T switched-line phase shifter and (b) measured performance from DC-18   GHz

(Tan et al., 2003). See the colour plate.

In contrast with switched network phase shifters, the distributed MEMS manual-line design consists of a single line that has been periodically loaded with MEMS variable or switched capacitors. This results in low loss phase shifters, particularly millimeter-wave bands. Figure 34 shows a photograph and a diagram of a iii-bit phase shifter fabricated on a glass substrate using MEMS switches and coplanar-waveguide lines (Hung, Dussopt, & Rebeiz 2004). By activating the capacitors, the total value of the loaded capacitance varies, which results in different stage shifts. The results prove an average loss of two.7   dB at 78   GHz (0.ix   dB/bit) with the reflection loss of beneath ten   dB over all eight states.

Figure 34. (a) Photograph and (b) diagram of the three-bit W-ring stage shifter.

(Hung, Dussopt, &amp; Rebeiz 2004)

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Performance improvement methods

Shigeyuki Takano , in Thinking Machines, 2021

half dozen.five.three.ane Shifter representation

A multiplier tin be approximated through the utilise of a left-shifter in the case of a fixed-point or integer numerical representation. Nosotros can obtain an index of the approximated source operand $src2$ as follows.

(6.36) $src2\ge \arg \underset{i}{\max }(2^i)$

Thus, we can employ a left-shifter for multiplication $\mathrm{1000}u\mathrm{50}t(⁎)$ as follows.

(6.37) $\mathrm{Thousand}ult(src1,src2)\le \min (MA\mathrm{Ten},src1\ll i)$

where MAX is the available maximum value in the numerical representation. Thus, the maximum error is approximately $MAX/2$ .

A simple multiplier for M-$.25, × N-bits, consists of an $\mathrm{Chiliad}\times N$ AND gate array and Due north ( $M+1$ )-scrap adders. Then, an NAND2 equivalent number of gates for the multiplier $A_{gu\mathrm{50}t}$ is as follows:

(6.38) $A_{\mathrm{grand}ult}=\mathrm{one.5}\times M\mathrm{Northward}+4\times (M+\mathrm{i})N$

where an AND gate and a single-bit full-adder take a i.5 and 4.0 NAND2 equivalent numbers of gates.

The shifter can consist of M two-operand 1-bit multiplexers on one phase and on Due north stages. An NAND2 equivalent number of gates for the shifter $A_{shift}$ is as follows.

(6.39) $A_{shift}=\mathrm{three.five}\times {\log }_{2}M\times {\log }_{\mathrm{ii}}\mathrm{Chiliad}$

The maximum shift corporeality N is equal to ${\log }_{\mathrm{ii}}K$ , and the shifter has ${\log }_{\mathrm{two}}M$ levels of selectors at most. Therefore, the expanse ratio R tin be represented as follows.

(6.twoscore) $R=\frac{A_{mult}}{A_{\mathrm{southward}hift}}=\frac{5.5M^2(1+\mathrm{iv.0}/(\mathrm{v.five}M))}{\mathrm{3.v}{({\log }_{\mathrm{two}}M)}^2}>\frac{5.5M^2}{\mathrm{iii.5}{({\log }_{\mathrm{ii}}M)}^2}$

in which information technology is causeless that Northward is equal to K. Fig. vi.36(a) shows the expanse ratio from the shifter approximation to a simple multiplier in terms of the equivalent number of NAND2 gates. The multiplier can easily increase its area by the multiplicand N. A shifter approximation has at least a 5-times smaller area, and a 6-times smaller area for a 16-chip datapath, as shown in Fig. half dozen.36(b).

Figure half-dozen.36. Advantage of shifter-based multiplication in terms of area.

Fig. half-dozen.37 shows a multiplier-free compages [350]. The authors proposed an algorithm to catechumen from a floating-point to a fixed-point numerical representation consisting of three phases. The first phase quantizes to 8-bits until achieving convergence of the parameter updates after the network model training. The second phase uses a student-teacher learning to improve the inference accurateness, which is degraded by the quantization. The last phase uses an ensemble learning in which multiple training is conducted for the aforementioned network compages and model ensembles.

Figure half-dozen.37. Multiplier-gratuitous convolution compages and its inference functioning [350].

Fig. 6.37(a) shows a processing node with a shifter on the beginning pipeline stage replacing the need for multipliers. This architecture uses an adder-tree method to accrue the ready of multiplications. Fig. six.37(b) shows the mistake rate on the ImageNet 2012 classification trouble. There is gap between the floating-signal and the proposed fixed-bespeak errors; however, preparation with stage-1 decreases this gap, and stage-2 obtains a lower error rate with a similar range as the floating-point approach.

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Recent Advances in Magnetic Insulators – From Spintronics to Microwave Applications

Boris A. Kalinikos , Alexey B. Ustinov , in Solid State Physics, 2013

4.1 Nonlinear Stage Shifters

Let united states of america consider the model of the YIG-film nonlinear stage shifter shown in Fig. 7.xviii. More often than not, a complex transmission coefficient of the device can be written every bit

Figure vii.18. Diagram of the nonlinear phase shifter.

(seven.39) $\dot{H}\left(\omega ,P_{\mathrm{in}}\right)=\sqrt{H_{\mathrm{p}}\left(\omega ,P_{\mathrm{in}}\right)}exp\left[\mathrm{i}\phi \left(\omega ,P_{\mathrm{in}}\right)\right],$

where H _p(ω,P _in) is the ability transmission coefficient and φ(ω,P _in) is the stage difference between input and output signals, that is, the stage-frequency characteristic of the device. The foursquare root of H _p(ω,P _in) is the AFC of the nonlinear phase shifter. It is worth noting that both characteristics are functions of the input power P _in. The insertion loss L (in decibels) can exist calculated as 50  =   10 log(H _p).

The power transmission coefficient for a spin-wave device is determined by the losses associated with the excitation and reception of the SW past microstrip antennas, also as past the losses due to propagation of the SW in the ferrite moving picture. Therefore, a formula for the full power manual coefficient of the nonlinear phase shifter can exist written in the form

(7.40) $H_{\mathrm{p}}\left(\omega ,P_{\mathrm{in}}\right)=H_{\mathrm{exc}}\left(\omega \right)H_{\mathrm{SW}}\left(\omega ,P_{\mathrm{in}}\right)H_{\mathrm{rec}}\left(\omega \right)$

where H _exc and H _rec are the coefficients characterizing efficiency of the SW excitation and reception, respectively, and H _SW is the coefficient characterizing attenuation of the SW propagating in the ferrite moving picture.

A formula for the total stage shift of the microwave point passed through the nonlinear phase shifter can be written as

(7.41) $\phi \left(\omega ,P_{\mathrm{in}}\right)={\phi }_{\mathrm{exc}}\left(\omega \right)+{\phi }_{\mathrm{SW}}\left(\omega ,P_{\mathrm{in}}\right)+{\phi }_{\mathrm{rec}}\left(\omega \right)$

where φ _exc and φ _rec are the phase shifts of the signal accumulated during the SW excitation and reception processes, respectively, and φ _SW is the phase shift accumulated during the propagation of the spin wave in the ferrite motion-picture show.

For relatively thick films (thicknesses greater than 10   μm), the coefficients H _exc and H _rec can be calculated using the linear SW excitation theory [xi], while H _SW and φ _SW can be calculated using Eqs. (7.thirty)–(seven.32) and (vii.37) and (7.38). In order to establish a connexion between P _in and |u|ⁱⁱ, a coefficient A  = P _in/|u|² tin can be calculated with the spin-wave theory [xi].

Figures 7.nineteen and 7.20 show operational characteristics of the nonlinear phase shifter based on surface SW. The device prototype was made with two 50   μm wide and 2   mm long brusk circuited microstrip antennas evaporated onto a grounded alumina substrate of 500   μm thickness. The distance between the antennas d was 4.6   mm. The antennas were fed by microstrip manual lines of 50   Ω characteristic impedance. A 13.half dozen   μm thick, 2   mm wide, and forty   mm long YIG unmarried-crystal film strip was utilized in the nonlinear stage shifter. The film was grown past liquid-stage epitaxy on 500   μm thick GGG substrate. The YIG film demonstrated a narrow ferromagnetic resonance line-width ΔH of 0.5   Oe at a frequency of 5   GHz and a saturation magnetization 4πM ₀ of 1947   G. The YIG/GGG strip was positioned, with the YIG side down, over the microstrip antennas.

Figure 7.19. Return loss versus frequency characteristic (a) and amplitude-frequency characteristics (b) of the nonlinear phase shifter measured for the different input power values P _in equally indicated.

Figure 7.xx. Insertion loss (a) and differential nonlinear phase shift (b) measured and calculated for the different frequencies as indicated. The symbols and the solid curves show experimental and theoretical data, respectively.

Figure 7.nineteen shows a typical return loss versus frequency feature and an AFC of the nonlinear phase shifter measured for the bias magnetic field H  =   1431   Oe for the different input power levels P _in in the range from −   6 to 23   dBm. For the pocket-size input power of −   half dozen   dBm, the SW propagated in a linear regime within the entire operating frequency range. During the increase in P _in, the render loss was varied by negligibly small values, whereas the insertion loss was considerably increased. In particular, the minimum insertion loss of −   7   dB observed effectually the frequency of 6.34   GHz was increased up to −   15   dB with the increment in the input power upwards to 23   dBm. Such a behavior of the AFC was due to the nonlinear damping of the carrier SW.

Figure 7.20 show the dependences of the insertion loss and the differential nonlinear stage shift Δφ_NL from the input power P _in measured for the different frequencies of the input signal. As is clear from the figure, besides the insertion loss increasing, the differential nonlinear stage shift Δφ_NL occurs with the increase in P _in. The measured value of Δφ_NL depended on the input indicate frequency. In particular, the increment in the frequency led to the increase in Δφ_NL up to the values of more 180 degrees (see Fig. 7.20b) that is important for the stage shifter applications (see Section 4.2).

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Floating-Signal Representation, Algorithms, and Implementations

Miloš D. Ercegovac , Tomás Lang , in Digital Arithmetics, 2004

Double-Datapath Implementation

In the single-datapath implementation the critical path includes two variable shifters: one for alignment of the operands and the other for normalization of the consequence. However, as indicated before, the normalization of the outcome requires a shift of more than than one position but when the performance is subtraction and the exponent deviation is goose egg or one; moreover, for this case the alignment is at most of one position. Consequently, every bit shown in Effigy 8.9, it is possible to define two disjoint paths:

Figure eight.9. Double-path implementation of floating-betoken addition.

•

CLOSE, for subtraction and exponent difference of cipher or one

•

FAR, for addition and for subtraction with exponent difference larger than one

In the Close path, at that place is a simple shifter for alignment of at most one position, the adder, the variable left shifter for normalization, and the module for rounding. ¹⁸ In contrast, the FAR path has a variable right shifter for alignment, the adder, a 1-position left-correct shifter for normalization, and the module for rounding.

The dependence graph of the double-path scheme (significand part only) is shown in Figure 8.10.

FIGURE 8.10. Dependence graph for double-path scheme.

To residue the delay of both paths, the following has to exist considered:

1.

To reach a higher throughput, the floating-point adder is pipelined, as shown in Effigy 8.11. As tin be seen, to pipeline the double-datapath implementation it is necessary to have 2 adders, one per path, because the add-on occurs in different stages of the pipeline: in the CLOSE path simultaneously with the variable right shifter of the FAR path, and in the FAR path simultaneously with the variable left shifter of the Shut path.

FIGURE viii.11. Pipelined implementations: (a) Single-path scheme. (b) Double-path scheme.

2.

To reduce the latency, the rounding is combined with the adder and performed earlier the normalization. This combined addition + rounding is performed by having a compound adder (which produces the sum and the sum plus 1) and the correct rounded result is selected from the two possible outputs. Specifically:

• For the CLOSE path, roundup might be required when the exponent divergence is 1 and the output of the adder is normalized.

EXAMPLE 8.half dozen

$\begin{array}{l}\text{SUB}\frac{\begin{array}{l}\text{1}\text{.1100100}\\ \text{0}\text{.10000001}\end{array}}{1.01000111}\\ \begin{array}{cc}\text{Round}& 1.0100100\end{array}\end{array}$

• For the FAR path, roundup might be required both when the issue of addition is normalized or unnormalized (with 1 additional integer chip for addition and with one leading zero for subtraction). Since the rounding is washed before normalization, the selection of the right output has to take into account the various positions of the rounding chip.

To illustrate the functioning of the ADD, Round, and NORMALIZE module nosotros prove the instance of constructive add-on and rounding to nearest. ^xix In this case, the Add together role produces 2 outputs: Sum (of inputs) and Sum + i (sum-plus-ane), both upward to bit position one thousand − 1 (L). Moreover, we accept too $.25 m (G), m + 1 (R), and the sticky flake (T), which stand for to the operand that has been shifted right. Then, for the choice amidst the two outputs of the adder, two situations have to be considered:– The result of addition (Sum) is normalized. In this situation, the rounded result is (L is scrap position one thousand − 1 of Sum)

8.51 $rounded=\{\begin{array}{cc}Su\mathrm{thou}& if\left(G^{\prime }+{\mathrm{50}}^{\prime }{R}^{\prime }{T}^{\prime }\right)=1\\ \mathrm{South}u\mathrm{one\; thousand}& ifK\left(R+T+\mathrm{Fifty}\right)=1\end{array}$

– The result of addition (Sum) has one additional integer fleck. In this situation, the rounded issue is (L ^* is position yard − two of Sum)

8.52 $round\mathrm{due\; east}d=\{\begin{array}{cc}R1\mathrm{Due\; south}HIFT(Sum)& if\left({\mathrm{Fifty}}^{\prime }+(L^{\ast })\prime G^{\prime }{R}^{\prime }{T}^{\prime }\right)=1\\ R1\mathrm{South}HIFT(Su\mathrm{one\; thousand}+o\mathrm{north}e)& ifL\left({\mathrm{Fifty}}^{\ast }+G+R+T\right)=\mathrm{one}\end{array}$

3.

The magnitude subtraction is performed with a 2's complement adder, equally discussed for the single-datapath case. However, for the instance in which the exponents are the aforementioned (Close path) it is non possible to perform a comparison of the significants earlier the adder, since this adder is in the first stage of the pipeline. Consequently, to avert the 2'south complement of the event when it is negative (which would require an incrementer), we do every bit follows:

•

Flake-invert one operand.

•

Select the sum plus one output if the result is positive and the chip invert of the sum if the result is negative.

Annotation that considering of the swap at the input, the difference can be negative just when the exponents are equal, and therefore this situation does not disharmonize with the roundup.

In the FAR datapath, the bandy assures that the output is always positive.

four.

Leading-zeros apprehension (LZA) is included in the CLOSE path (every bit discussed for the single-datapath case).

As seen in Effigy viii.11, the utilise of a double-path implementation might reduce the latency past ane pipeline stage. However, it increases significantly the area.

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Function Evaluation

Miloš D. Ercegovac , Tomás Lang , in Digital Arithmetic, 2004

Implementation

As shown in Effigy 10.18 , the overall implementation requires two variable shifters, one [4:2] adder, ane [3:2] adder, one CPA, the selection function module, two multiplexers, a module with a table for generating L_j constants, and four registers.

FIGURE 10.18. Implementation of radix-2 algorithm for computing ln(10).

The delay is

x.76 $T_{\mathrm{Fifty}N}=[\max ((\max (t_{\mathrm{southward}e\mathrm{50}},t_{\mathrm{south}hift})+t_{4-2}),(t_{sel}+t_{table}+t_{CSA})+t_{RE\mathrm{1000}}]\mathrm{thousand}+t_{CPA}$

where k is the number of iterations. The table L contains 1000/2 × 2 constants. The admission to the table tin exist removed from the critical path (see Practise ten.xviii).

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Jakobson, Roman (1896–1982)

L.R. Waugh , in International Encyclopedia of the Social & Behavioral Sciences, 2001

7 Grammar

Jakobson also focused on the function of grammatical categories. His famous article on Shifters (1957) focuses on those elements whose general significant in the code tin only be specified by taking into business relationship their employ in messages, because the codal meaning includes data about particular elements of the spoken communication event. For example, deictic pronouns designate speaker (I) and addressee (you). Thus, language encodes pragmatic factors of the context of utterance and linguistics necessarily includes syntax, semantics, and pragmatics (meet Deixis ; Pragmatics: Linguistic ).

Grammatical categories (both morphological and syntactic), for Jakobson, are defined as those which are obligatorily present in the construction of acceptable messages (1959a, 1959b), whereas particular lexical categories (e.g., words referring to space) are optional. Through this view of grammar, Jakobson provided a semantic and operational arroyo to the relation betwixt linguistic communication and knowledge: grammatical categorizations provide the necessary patterns of idea.

Jakobson was too inspired by Charles Sanders Peirce's notion that the essence of a sign is its interpretation, that is, its translation by some further sign. Henceforth, he defined the signatum equally that which is interpretable or, better, translatable (Jakobson 1959b) by a potentially unlimited series of signs (run across Interpretation and Translation: Philosophical Aspects ). He characterized the Peircian arroyo as the only sound footing for a strictly linguistic semantics.

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Managing Modify

John J. Fay , David Patterson , in Contemporary Security Management (Fourth Edition), 2018

Blame Shifters

The people most afflicted by change on a personal level seem in many cases to be blame-shifters. The usual laments include, "The boss had it in for me. The company deceived me. Lady Luck was confronting me." They are cocky-doubters who end up failing because they fear success. They tend to undervalue their talents and destroy their chances of getting ahead. The losers in change are sometimes people who are then locked into their own identity they cannot tolerate working differently. Change to the job is unacceptable because the task and the person are ane and the same. "They wanted me to be the fleet manager when I was already the head of security. They can't practice that to me." But they did.

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