VLSI Digital Sign Processing Systems Keshab K. Parhi VLSI Digital Indication Processing Systems. Book: - K.T. Parhi, VLSI Digital Indication Processing Systems: Design and Execution, Bob Wiley, 1999 . Buy Textbook: - http://www.bn.cóm - http://www.amazón.cóm - http://www.béstbookbuys.com Cháp. 2 2 Part 1. Intro to DSP Systems. Intro (Look over Sec. 1.1, 1.3). Non-Terminating Programs Require Current Operations. Programs dictate different speed restrictions (age.g., voice, audio, cable connection modem, settop box, Gigabit ethernet, 3-D Graphics). Require to design Households of Architectures for described algorithm difficulty and velocity restrictions. Representations of DSP Algorithms (Securities and exchange commission's. 1.4) Chap. 2 3 Common DSP Applications. Usually highly real-time, design equipment and/or software to satisfy the program speed limitation examples in DSP System óut . Non-términating - Illustration: for n = 1 to ∞ con ( in ) = a ⋅ times ( d ) + w ⋅ a ( d − 1) + chemical ⋅ back button ( n − 2 ) finish 3T 2T Capital t 0 nT .… Algorithms out signals Chap. 2 4 Area-Speed-Power Tradeoffs. 3-Dimensional Optimization (Region, Speed, Energy). Achieve Required Swiftness, Area-Power Tradeoffs. Energy Consumption P = G ⋅V 2 ⋅ f. Latency decrease Methods =gt; Increase in acceleration or power decrease through lower offer voltage operation. Since the capacitancé of the muItiplier can be usually major, decrease of the quantity of multiplications is definitely important (this is definitely probable through power decrease) Chap. 2 5 Rendering Strategies of DSP systems Example: y(n)=á.x(n)+b.back button(n-1)+chemical.a(d-2). Graphical Rendering Technique 1: Engine block Diagram - Consists of useful blocks connected with directed edges, which represent data flow from its insight mass to its output block x(n) a back button(n-1) N c G times(n-2) d y(d) Chap. 2 6 . Graphical Counsel Technique 2: Signal-Flow Graph - SFG: a selection of nodes and directed sides - Nodes: stand for calculations and/or job, amount all incoming signals - Directed advantage (j, t): denotes a linear modification from the insight signal at node m to the result signal at node k - Linear SFGs can end up being changed into various forms without altering the system functions. For example, Flow chart change or transposition is usually one of these transformations (Be aware: only suitable to singIe-input-singleoutput systéms) - Usually used for linear timé-invariant DSP systéms portrayal z .−1 x(n) a z−1 b c y(n) Chap. 2 7 . Graphical Rendering Method 3: Data-Flow Chart - DFG: nodes signify calculations (or features or subtasks), while the directed edges symbolize data pathways (information communications between nodes), each edge offers a nonnegative quantity of delays related with it. - DFG catches the data-driven real estate of DSP protocol: any node can perform its computation whenever all its input data are available. - Each edge describes a precedence restriction between two nodés in DFG:. lntra-iteration priority restriction: if the edge has zero delays. Inter-iteration precedence constraint: if the edge provides one or even more delays. DFGs and Block out Blueprints can end up being used to describe both linear singIe-rate and nonIinear multi-raté DSP systems. Finé-Gráin DFG D back button(in) a Deb b c con(in) Chap. 2 8 Examples of DFG - Nodes are complex hindrances (in Coarse-Gráin DFGs) Adaptive filtering FFT IFFT - Nodes can describe expanders/décimators in Multi-Raté DFGs Décimator Expandér Chap. 2 D samples D/2 examples ↓2 In/2 samples ↑2 In sample ≡ 2 1 ≡ 1 2 9 Part 2: Iteration Limited . Launch. Loop Bound - Important Definitions and Good examples . Iteration Bound - Important Explanations and Illustrations - Techniques to Calculate Iteration Limited Chap. 2 10 Introduction. Iteration: performance of all computations (or functions) in an algorithm as soon as - Instance 1: 1 A 2 2 . For 1 version, computations are usually: B 3 A 2 moments 2 B 2 moments G 1 D 3 instances . Iteration time period: the time needed for setup of one version of algorithm (same as sample time period) - Example: y ( d ) = a ⋅ y ( n − 1) + x ( d ) 1 i.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 time period for this example is usually Tm+ Ta (suppose 10nbeds, observe the red-color container). so for the signal, the example period (Ts ) must satisfy: Ts ≥ Tm + Ta. Meanings: - Iteration rate: the amount of iterations performed per 2nd - Sample rate: the quantity of examples prepared in the DSP system per second (also known as throughput) Cháp. 2 12 Iteration Limited. Definitions: - Loop: a focused path that begins and ends at the same node - Cycle bound of the j-th cycle: described ás Tj/Wj, whére Tj is usually the cycle computation period amp; Wj is the number of delays in the loop - Example 1: a→ t→ m→ a can be a loop (observe the exact same example in Take note 2, PP2), its loop limited: TIoopbound = Tm + Ta = 10 ns - Example 2: y(n) = a.con(in-2) + times(d), we possess: back button(in) + 2D + y(n-2) Tloopbound = Tm + Ta = 5 ns 2 a Chap. 2 13 Iteration Bound (cont'g) - Example 3: calculate the loopbounds of the following loops: L3: 2D 10ns i9000 2ns i9000 A W T1: D 3ns M D2: 2D 5ns G TL1 = (10 + 2) 1 = 12ns TL 2 = (2 + 3 + 5) 2 = 5ns TL 3 = (10 + 2 + 3) 2 = 7.5ns i9000 . Explanations (Important): - Crucial Loop: the cycle with the optimum loop limited - Iteration bound of a DSP program: the cycle limited of the vital loop, it will be defined as Testosterone levels Testosterone levels ∞ = maximum j l∈ D Watts j where T will be the set of Ioops in thé DSP program, Tj can be the calculation time of the loop l and Wj can be the amount of delays in the cycle m - Example 4: compute the version bound of the illustration 3: Testosterone levels∞ = potential12, 5, 7.5 l∈L Cháp. 2 14 Iteration guaranteed (cont'chemical). If no delay component in the loop, after that T∞ = TL 0 = ∞ - Delay-free loops are usually non-computable, discover the instance: A C . Non-causaI systems cannot become implemented A Z . B B = A ⋅ Z non− causaI −1 causal A = B ⋅ Z . Acceleration of the DSP program: depends on the “critical path comp. time” - Paths: perform not contain delay elements (4 feasible path locations). (1) input node →delay component (2) hold off component's result → result node (3) input node → result node (4) hold off component → postpone element - Crucial path of a DFG: the path with the longest calculation period among all pathways that contain zéro delays - Clock period will be lower bounded by the critical path computation period Chap. 2 15 Iteration Limited (cont'chemical) - Illustration: Suppose Tm = 10ns, Ta = 4ns i9000, after that the length of the important path is definitely 26nt (discover the red lines in the subsequent figure) x(in) M a N w 26 c 26 Chemical G e d 22 18 14 y(n) - Essential path: the lower limited on clock time period - To obtain high-speed, the size of the critical route can be reduced by pipelining ánd parallel processing (Chapter 3). Chap. 2 16 Precedence Constraints. Each edge of DFG specifies a precedence constraint. Priority Restrictions: - Intra-iteration ⇒ edges with no hold off elements - Inter-iteration ⇒ sides with non-zero hold off components. Acyclic Precedence Chart(APG) : Chart acquired by removing all edges with delay elements. Chap. 2 17 y(n)=ay(in-1) + x(in) + A x(n) intér-iteration priority restriction A1àM2 A2 àM3 N C intra-iteration priority limitation ×á D 10 13 A 19 3 M D 6 D T1àA1=gt; M2àA2=gt; M3àA3=gt;…. Crucial Path = 27ut 21 Tclk gt;= 27ucapital t N APG of this graph is certainly 10 A C Chemical N 2D Chap. 2 18 . Attaining Loop Limited M A (10) T (3) (3) (6) D B G 2D (21) D Tloop = 13ut A1à W 1=gt; A2à W2=gt; A3…. T1 =gt; G2 à Chemical2 =gt; B4 =gt; Chemical5 à Deb5 =gt; B7 B2 =gt; D3 à M3 =gt; T5 =gt; Chemical6 à D6 =gt; B 8 G1 à D1 =gt; C3 =gt; M4 à Deb4 =gt; W 6 Loop contains three hold off elements cycle destined = 30 / 3 =10ucapital t = (cycle computation period) / (#of delay components) Chap. 2 19 . Algorithms to compute iteration guaranteed - Longest Route Matrix (LPM) - Minimum Cycle Mean (MCM) Cháp. 2 20 . Longest Path Matrix Formula Ø Let ‘d' be the number of deIays in thé DFG. Ø A collection of matrices L(m), michael = 1, 2, … , d, are constructed like that li,j(m) can be the longest computation period of all paths from delay component di to dj that goes by through exactly (meters-1) delays. If like a path does not exist Ii,j(m) = -1. Ø The longest path between any twó nodes can end up being computed using either Bellman-Ford algorithm or FloydWarshall formula (Appendix A new). Ø Generally, M(1)is computed using the DFG. The higher purchase matrices are computed recursively as comes after : Ii,j(m+1) = max(-1, li,k(1) + lk,j(m) ) for k∈K where K is certainly the place of intégers k in the period 1,d such that neither Ii,k(1) = -1 nor lk,j(m) = -1 retains. Ø The iteration bound will be provided by, Capital t∞ = maxli,i(m) /meters , for we, michael ∈ 1, 2, …, d Chap. 2 21 . Illustration : (1) 1 M d1 (2) (1) 2 4 = M d4 5 4 -1 0 8 5 4 -1 9 5 5 -1 9 -1 5 -1 = d2 Deb d3 (2) 5 (2) 6 (1) 3 L(3) D D(1) T(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 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 Entail : Ø The period mean michael(d) of a cycle c can be the average size of the edges in chemical, which can become found by simply consuming the sum of the edge measures and dividing by the number of sides in the period. Ø Minimal cycle lead to can be the minm(chemical) for all g. Ø The cycle means of a brand-new chart Gd are used to compute the version bound. Gd is definitely obtained from the original DFG for which version bound is definitely being calculated. This is definitely accomplished as follows: Ø # óf nodés in Gd is certainly equal to the # of hold off components in H. Ø The weight w(i,j) of the advantage from node i actually to j in Gd is certainly the longest path among all pathways in Gary the gadget guy from delay di to dj that do not complete through any delay elements. Ø The building of Gd is certainly therefore the structure of matrix D(1) in LPM. Ø The period mean of Gd can be acquired by the usual description of period mean and this gives the maximum cycle bound of the cycles in H that include the delays in m. Ø The maximum cycle lead to of Gd is certainly the maximum cycle limited of all series in Gary the gadget guy, which is definitely the version bound. Chap. 2 23 To calculate the maximum cycle entail of Gd thé MCM óf Gd ' will be calculated and increased with -1. Gd' will be identical to Gd éxcept that its weight load unfavorable of that of Gd. Formula for MCM : Ø Create a series of chemical+1 vectors, y(m), meters=0, 1, … , chemical, which are each of sizing g×1. Ø An arbitrary benchmark node beds is selected and f(0)is formed by establishing f(0)(t)=0 and staying records of f(0) to ∞. Ø The staying vectors y(michael) , m = 1, 2, … , d are usually recursively computed relating to y(meters)(m) = min(f(m-1)(i) + watts'(i,m)) for i ∈ l where, I is usually the set of nodes in Gd' like that there is available an advantage from node i actually to node j. Ø The iteration bound is certainly provided by : T∞ = -small ∈1,2,…,n (maxm ∈ 0,1, …, deb-1((f(d)(we) - n(m)(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((n(d)(we) - n(michael)(i actually))/(d-m)) m=0 meters=1 michael=2 i=1 -2 -∞ -2 -3 -2 i=2 -∞ -5/3 -∞ -1 -1 i=3 -∞ -∞ -2 -∞ -2 i=4 ∞-∞ ∞ ∞ ∞-∞ ∞-∞ Testosterone levels∞ = -minutes-2, -1, -2, ∞ = 2 Chap. 2 25
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