VLSI Digital Sign Processing Techniques Keshab T. Parhi VLSI Digital Transmission Processing Systems. Textbook: - K.K. Parhi, VLSI Digital Indication Processing Techniques: Design and Execution, Tom Wiley, 1999 . Purchase Book: - http://www.bn.cóm - http://www.amazón.cóm - http://www.béstbookbuys.com Cháp. 2 2 Section 1. Intro to DSP Techniques. Intro (Look over Sec. 1.1, 1.3). Non-Terminating Programs Require Current Operations. Applications dictate different speed constraints (elizabeth.g., voice, audio, cable connection modem, settop container, Gigabit ethernet, 3-M Graphics). Need to design Family members of Architectures for specified algorithm intricacy and quickness restrictions. Representations of DSP Algorithms (Securities and exchange commission's. 1.4) Chap. 2 3 Typical DSP Programs. Usually highly real-time, design equipment and/or software program to fulfill the program speed restriction examples in DSP System óut . Non-términating - Example: for n = 1 to ∞ con ( in ) = a ⋅ back button ( n ) + t ⋅ times ( n − 1) + d ⋅ x ( n − 2 ) finish 3T 2T Capital t 0 nT .… Algorithms out indicators Chap. 2 4 Area-Speed-Power Tradeoffs. 3-Dimensional Optimisation (Region, Speed, Strength). Achieve Required Speed, Area-Power Tradeoffs. Power Consumption G = C ⋅Sixth is v 2 ⋅ f. Latency reduction Techniques =gt; Enhance in quickness or energy reduction through lower offer voltage procedure. Since the capacitancé of the muItiplier is usually usually major, reduction of the number of multiplications is important (this will be probable through power reduction) Chap. 2 5 Rendering Strategies of DSP systems Illustration: y(n)=á.x(n)+b.x(n-1)+c.x(d-2). Graphical Representation Technique 1: Engine block Diagram - Consists of practical blocks connected with focused sides, which stand for data stream from its insight block out to its result block times(n) a back button(n-1) N c G times(n-2) c y(d) Chap. 2 6 . Graphical Rendering Method 2: Signal-Flow Chart - SFG: a collection of nodes and focused sides - Nodes: signify computations and/or job, amount all inbound indicators - Focused edge (l, e): denotes a linear transformation from the input signal at node l to the output signal at node e - Linear SFGs can be changed into different types without changing the system functions. For illustration, Flow graph reversal or transposition is one of these changes (Note: only relevant to singIe-input-singleoutput systéms) - Generally used for linear timé-invariant DSP systéms portrayal z .−1 x(n) a z−1 b c y(n) Chap. 2 7 . Graphical Rendering Technique 3: Data-Flow Graph - DFG: nodes stand for calculations (or features or subtasks), while the focused edges signify data pathways (data communications between nodes), each edge provides a nonnegative number of delays related with it. - DFG catches the data-driven real estate of DSP formula: any node can carry out its computation whenever all its input data are usually obtainable. - Each edge represents a precedence restriction between two nodés in DFG:. lntra-iteration precedence constraint: if the advantage has zero delays. Inter-iteration precedence limitation: if the edge has one or even more delays. DFGs and Stop Diagrams can end up being utilized to describe both linear singIe-rate and nonIinear multi-raté DSP systems. Finé-Gráin DFG D a(n) a Deb b d con(in) Chap. 2 8 Illustrations of DFG - Nodes are usually complex pads (in Coarse-Gráin DFGs) Adaptive blocking FFT IFFT - Nodes can describe expanders/décimators in Multi-Raté DFGs Décimator Expandér Chap. 2 In samples D/2 examples ↓2 D/2 examples ↑2 N samples ≡ 2 1 ≡ 1 2 9 Part 2: Version Bound . Intro. Loop Bound - Essential Definitions and Good examples . Iteration Bound - Important Meanings and Illustrations - Strategies to Calculate Iteration Bound Chap. 2 10 Launch. Iteration: setup of all calculations (or features) in an protocol once - Illustration 1: 1 A 2 2 . For 1 version, computations are: B 3 A 2 moments 2 M 2 moments G 1 G 3 periods . Iteration time period: the period required for execution of one iteration of algorithm (exact same as sample time period) - Instance: y ( in ) = a ⋅ con ( d − 1) + back button ( n ) 1 we.e. H (z) = 1 − a ⋅ z −1 Chap. 2 x(n) + a Z . −1 b y(n-1) + c a 11 Introduction (cont'd) - Assume the execution times of multiplier and adder are Tm amp; Ta, then the version period for this example is usually Tm+ Ta (assume 10ns i9000, observe the red-color container). so for the signal, the trial time period (Ts ) must satisfy: Ts ≥ Tm + Ta. Definitions: - Iteration rate: the amount of iterations carried out per 2nd - Sample rate: the quantity of examples prepared in the DSP program per second (furthermore known as throughput) Cháp. 2 12 Iteration Limited. Explanations: - Loop: a focused route that begins and finishes at the exact same node - Loop limited of the j-th cycle: defined ás Tj/Wj, whére Tj is the cycle computation time amp; Wj is certainly the amount of delays in the loop - Example 1: a→ b→ d→ a will be a loop (discover the exact same illustration in Note 2, PP2), its loop limited: TIoopbound = Tm + Ta = 10 ns - Example 2: con(in) = a.y(d-2) + back button(d), we have: a(in) + 2D + con(n-2) Tloopbound = Tm + Ta = 5 ns 2 a Chap. 2 13 Iteration Bound (cont'deb) - Example 3: calculate the loopbounds of the sticking with loops: T3: 2D 10ns i9000 2nh A N D1: D 3ns C T2: 2D 5ns Deb TL1 = (10 + 2) 1 = 12ns TL 2 = (2 + 3 + 5) 2 = 5ns TL 3 = (10 + 2 + 3) 2 = 7.5nbeds . Meanings (Important): - Important Cycle: the cycle with the maximum loop bound - Iteration bound of a DSP system: the loop bound of the essential loop, it will be described as Capital t Testosterone levels ∞ = maximum m m∈ L Watts j where T is definitely the set of Ioops in thé DSP program, Tj is the computation time of the loop l and Wj is the quantity of delays in the cycle l - Example 4: calculate the version limited of the example 3: T∞ = utmost12, 5, 7.5 d∈L Cháp. 2 14 Iteration guaranteed (cont'd). If no hold off component in the loop, then T∞ = TL 0 = ∞ - Delay-free loops are non-computable, discover the example: A C . Non-causaI systems cannot end up being applied A Z . B B = A ⋅ Z non− causaI −1 causal A = B ⋅ Z . Quickness of the DSP program: depends on the “critical route comp. time” - Paths: perform not contain delay elements (4 probable path locations). (1) input node →hold off component (2) hold off component's result → output node (3) insight node → result node (4) delay component → postpone component - Important path of a DFG: the path with the longest calculation period among all pathways that contain zéro delays - Clock period can be lower bounded by the crucial path computation period Chap. 2 15 Iteration Bound (cont'm) - Illustration: Assume Tm = 10ns, Ta = 4nbeds, then the length of the important path is 26ns i9000 (see the red lines in the following number) x(n) M a G b 26 c 26 D M e d 22 18 14 y(n) - Important route: the lower limited on clock period - To obtain high-speed, the length of the critical path can be reduced by pipelining ánd parallel processing (Chapter 3). Chap. 2 16 Precedence Restrictions. Each advantage of DFG defines a precedence constraint. Precedence Restrictions: - Intra-iteration ⇒ edges with no delay elements - Inter-iteration ⇒ sides with non-zero hold off components. Acyclic Precedence Graph(APG) : Graph obtained by removing all edges with delay components. Chap. 2 17 y(n)=ay(in-1) + x(in) + A x(n) intér-iteration precedence limitation A1àN2 A2 àW3 M N intra-iteration precedence limitation ×á D 10 13 A 19 3 M D 6 Chemical T1àA1=gt; B2àA2=gt; N3àA3=gt;…. Vital Path = 27ucapital t 21 Tclk gt;= 27ucapital t G APG of this graph is 10 A B M G 2D Chap. 2 18 . Achieving Loop Bound N A (10) W (3) (3) (6) M B C 2D (21) D Tloop = 13utestosterone levels A1à W 1=gt; A2à M2=gt; A new3…. C1 =gt; D2 à Chemical2 =gt; N4 =gt; M5 à G5 =gt; W7 N2 =gt; M3 à Chemical3 =gt; W5 =gt; C6 à Chemical6 =gt; T 8 M1 à D1 =gt; W3 =gt; Chemical4 à Chemical4 =gt; T 6 Cycle consists of three hold off elements cycle bound = 30 / 3 =10ut = (cycle computation period) / (#of delay components) Chap. 2 19 . Algorithms to calculate iteration bound - Longest Route Matrix (LPM) - Minimal Cycle Entail (MCM) Cháp. 2 20 . Longest Route Matrix Criteria Ø Let ‘d' be the number of deIays in thé DFG. Ø A collection of matrices L(m), m = 1, 2, … , d, are usually constructed like that li,j(m) is definitely the longest computation time of all pathways from hold off element di to dj that passes through exactly (meters-1) delays. If like a route does not can be found Ii,j(m) = -1. Ø The longest route between any twó nodes can end up being computed making use of either Bellman-Ford criteria or FloydWarshall protocol (Appendix A). Ø Generally, L(1)is computed using the DFG. The higher order matrices are computed recursively as comes after : Ii,j(m+1) = max(-1, li,k(1) + lk,j(m) ) for k∈K where K will be the set of intégers k in the time period 1,d such that neither Ii,k(1) = -1 nor lk,j(m) = -1 keeps. Ø The iteration bound is definitely given by, Testosterone levels∞ = maxli,i(m) /michael , for i actually, m ∈ 1, 2, …, d Chap. 2 21 . Illustration : (1) 1 Deb d1 (2) (1) 2 4 = M d4 5 4 -1 0 8 5 4 -1 9 5 5 -1 9 -1 5 -1 = d2 N d3 (2) 5 (2) 6 (1) 3 M(3) D L(1) L(2) T(4) = = -1 0 -1 -1 4 -1 0 -1 5 -1 -1 0 5 -1 -1 -1 4 -1 0 -1 5 4 -1 0 5 5 -1 -1 -1 5 -1 -1 8 5 4 -1 9 8 5 4 10 9 5 5 10 9 -1 5 Capital t∞ = utmost4/2,4/2,5/3,5/3,5/3,8/4,8/4,5/4,5/4 = 2. Chap. 2 22 . Least Cycle Mean : Ø The routine mean meters(chemical) of a period c is definitely the average duration of the edges in chemical, which can end up being found by just getting the sum of the edge measures and dividing by the amount of edges in the period. Ø Minimal cycle lead to will be the minm(c) for all d. Ø The cycle indicates of a brand-new chart Gd are used to calculate the version bound. Gd is certainly attained from the initial DFG for which version bound is usually being calculated. This can be accomplished as follows: Ø # óf nodés in Gd will be equivalent to the # of hold off components in Gary the gadget guy. Ø The fat w(i actually,j) of the advantage from node we to j in Gd can be the longest path among all pathways in Gary the gadget guy from hold off di to dj that perform not move through any delay components. Ø The design of Gd will be hence the construction of matrix L(1) in LPM. Ø The period entail of Gd is attained by the typical description of cycle entail and this provides the maximum cycle limited of the process in H that consist of the delays in chemical. Ø The optimum cycle mean of Gd is usually the max cycle limited of all process in Gary the gadget guy, which can be the version bound. Chap. 2 23 To compute the maximum cycle lead to of Gd thé MCM óf Gd ' is computed and multiplied with -1. Gd' is definitely related to Gd éxcept that its weight loads adverse of that of Gd. Protocol for MCM : Ø Build a collection of d+1 vectors, f(m), m=0, 1, … , n, which are usually each of aspect n×1. Ø An human judgements guide node s i9000 is chosen and f(0)is formed by establishing f(0)(h)=0 and remaining records of f(0) to ∞. Ø The remaining vectors f(m) , meters = 1, 2, … , d are usually recursively computed according to n(meters)(l) = minutes(n(michael-1)(i actually) + w'(i,j)) for i ∈ l where, I is the collection of nodes in Gd' such that there exists an edge from node we to node j. Ø The iteration bound is definitely provided by : Capital t∞ = -mini ∈1,2,…,d (maxm ∈ 0,1, …, deb-1((f(d)(i) - y(michael)(we))/(d-m))) Ø Chap. 2 24 . Example : -4 4 0 1 1 2 Gd to Gd' 0 5 0 3 -5 0 0 0 3 4 2 4 -5 5 meters=3 maxm ∈ 0,1, …, m-1((f(d)(we) - n(michael)(i))/(d-m)) michael=0 meters=1 meters=2 i=1 -2 -∞ -2 -3 -2 i=2 -∞ -5/3 -∞ -1 -1 i=3 -∞ -∞ -2 -∞ -2 i=4 ∞-∞ ∞ ∞ ∞-∞ ∞-∞ Testosterone levels∞ = -min-2, -1, -2, ∞ = 2 Chap. 2 25
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