Review. EECS Components and Design Techniques for Digital Systems. Lec 26 CRCs, LFSRs (and a little power)

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1 EECS 50 - Components and esign Techniques for igital Systems Lec 6 CRCs, LFSRs (and a little power) avid Culler Electrical Engineering and Computer Sciences University of California, Berkeley Review Concept of error coding A a few etra bits (enlarges the space of values) that carry information about all the bits etect: Simple function to check of entire datacheck received correctly» Small subset of the space of possible values Correct: Algorithm for locating nearest valid symbol Hamming codes Selective use of parity functions istance # bit flips Parity: XOR of the bits => single error detection SECE» databitsp < p Outline Introduce LFSR as fancy counter Practice of Cyclic Redundancy Checks Burst errors in networks, disks, etc. Theory of LFSRs Power Linear Feedback Shift Registers (LFSRs) These are n-bit counters ehibiting pseudo-random behavior. Built from simple shift-registers with a small number of or gates. Used for: random number generation counters error checking and correction Advantages: very little hardware high speed operation Eample -bit LFSR: -bit LFSR Circuit counts through - different non-zero bit patterns. Left most bit determines shiftl or more comple operation Can build a similar circuit with any number of FFs, may need more or gates. In general, with n flip-flops, n - different non-zero bit patterns. (Intuitively, this is a counter that wraps around many times and in a strange way.) or or or or or or or Applications of LFSRs Performance: Can be used as a random number generator. In general, ors are only ever -input and never connect in series. Sequence is a pseudorandom sequence: Therefore the minimum clock period for these circuits is:» numbers appear in a random sequence T > T -input-or clock overhead» repeats every n - Very little latency, and independent of n! patterns This can be used as a fast counter, if the Random numbers useful in: particular sequence of count values is» computer graphics not important.» cryptography Eample: micro-code micro-pc» automatic testing Used for error detection and correction» CRC (cyclic redundancy codes)» ethernet uses them 6

Concept: Redundant Check Eample: TCP Checksum Send a message M and a check word C Simple function on <M,C> to determine if both received correctly (with high probability) Eample: XOR all the bytes in

*** bit i is XOR of ith bit of each byte 7 Application (HTTP,FTP, NS) Transport (TCP, UP) Network (IP) ata Link (Ethernet, 80.

sender, and included in a segment transmission.

2 Concept: Redundant Check Eample: TCP Checksum Send a message M and a check word C Simple function on <M,C> to determine if both received correctly (with high probability) Eample: XOR all the bytes in M and append the checksum byte, C, at the end Receiver XORs <M,C> What should result be? What errors are caught? *** bit i is XOR of ith bit of each byte 7 Application (HTTP,FTP, NS) Transport (TCP, UP) Network (IP) ata Link (Ethernet, 80.b) Physical TCP Packet Format TCP Checksum a 6-bit checksum, consisting of the one's complement of the one's complement sum of the contents of the TCP segment header and data, is computed by a sender, and included in a segment transmission. (note end-around carry) Summing all the words, including the checksum word, should yield zero Eample: Ethernet CRC- Application (HTTP,FTP, NS) Transport (TCP, UP) Network (IP) ata Link (Ethernet, 80.b) Physical CRC concept I have a msg polynomial M() of degree m We both have a generator poly G() of degree m Let r() = remainder of M() n / G() M() n = G()p() r() r() is of degree n What is (M() n r()) / G()? n bits of zero at the end tack on n bits of remainder So I send you M() n r() Instead of the zeros mn degree polynomial You divide by G() to check M() is just the m most signficant coefficients, r() the lower m n-bit Message is viewed as coefficients of n-degree polynomial over binary numbers 9 0 Announcements Reading XILINX IEEE 80. Cyclic Redundancy Check (pages -) ftp://ftp.rocksoft.com/papers/crc_v.tt Final on /5 What s Going on in EECS? Towards simulation of a igital Human Yelick: Simulation of the Human Heart Using the Immersed Boundary Method on Parallel Machines Galois Fields - the theory behind LFSRs LFSR circuits performs multiplication on a field. A field is defined as a set with the following: two operations defined on it:» aition and multiplication closed under these operations associative and distributive laws hold aitive and multiplicative identity elements aitive inverse for every element multiplicative inverse for every non-zero element Eample fields: set of rational numbers set of real numbers set of integers is not a field (why?) Finite fields are called Galois fields. Eample: Binary numbers 0, with XOR as aition and AN as multiplication. Called GF(). 0 = = 0 0- =? - =?

3 Galois Fields - The theory behind LFSRs Consider polynomials whose coefficients come from GF(). Each term of the form n is either present or absent. Eamples: 0,,,, and 7 6 = With aition and multiplication these form a field: A : XOR each element individually with no carry: Multiply : multiplying by n is like shifting to the left. So what about division (mod) X = with remainder 0 = with remainder 0 0 X Remainder Polynomial division CRC encoding serial_in serial_in When MSB is zero, just shift left, bringing in net bit When MSB is, XOR with divisor and shiftl Message sent: CRC decoding Galois Fields - The theory behind LFSRs serial_in These polynomials form a Galois (finite) field if we take the results of this multiplication modulo a prime polynomial p(). A prime polynomial is one that cannot be written as the product of two non-trivial polynomials q()r() Perform modulo operation by subtracting a (polynomial) multiple of p() from the result. If the multiple is, this corresponds to XOR-ing the result with p(). For any degree, there eists at least one prime polynomial. With it we can form GF( n ) Aitionally, Every Galois field has a primitive element, α, such that all non-zero elements of the field can be epressed as a power of α. By raising α to powers (modulo p()), all non-zero field elements can be formed. Certain choices of p() make the simple polynomial the primitive element. These polynomials are called primitive, and one eists for every degree. For eample, is primitive. So α = is a primitive element and successive powers of α will generate all non-zero elements of GF(6). Eample on net slide. 7 8

4 Galois Fields Primitives α 0 = α = α = α = α = α 5 = α 6 = α 7 = α 8 = α 9 = α 0 = α = α = α = α = α 5 = Note this pattern of coefficients matches the bits from our -bit LFSR eample. α = mod = or = In general finding primitive polynomials is difficult. Most people just look them up in a table, such as: 9 Primitive Polynomials Hardware shift left Galois Field Multiplication by Taking the result mod p() XOR-ing with the coefficients of p() when the most significant coefficient is. Obtaining all n - non-zero Shifting and XOR-ing n - times. elements by evaluating k for k =,, n - 0 Building an LFSR from a Primitive Poly For k-bit LFSR number the flip-flops with FF on the right. The feedback path comes from the output of the leftmost FF. Find the primitive polynomial of the form k. The 0 = term corresponds to connecting the feedback directly to the input of FF. Each term of the form n corresponds to connecting an or between FF n and n. -bit eample, uses FF s output or between FF and FF FF s input To build an 8-bit LFSR, use the primitive polynomial 8 and connect ors between FF and FF, FF and FF, and FF and FF5. Generating Polynomials CRC-6: G() = 6 5 detects single and double bit errors All errors with an o number of bits Burst errors of length 6 or less Most errors for longer bursts CRC-: G() = Used in ethernet Also bits of aed on front of the message» Initialize the LFSR to all s POWER Motivation Why should a digital designer care about power consumption? Portable devices: handhelds, laptops, phones, MP players, cameras, all need to run for etended periods on small batteries without recharging evices that need regular recharging or large heavy batteries will lose out to those that don t. Power consumption important even in tethered devices. System cost tracks power consumption:» power supplies, distribution, heat removal power conservation, environmental concerns In a span of 0 years we have gone from designing without concern for power consumption to (in many cases) designing with power consumption as the primary design constraint!

Battery Technology Battery technology has moved very slowly Moore s law does not seem to apply Li-Ion and NiMh still the dominate technologies Batteries still contribute significant to the weight of

5 Battery Technology Battery technology has moved very slowly Moore s law does not seem to apply Li-Ion and NiMh still the dominate technologies Batteries still contribute significant to the weight of mobile devices Nokia 6 - % Handspring PA - 0% Toshiba Portege 0 laptop - 0% 5 Basics Power supply provides energy for charging and discharging wires and transistor gates. The energy supplied is stored and dissipated as heat. Power: Rate of work being done w.r.t time. P dw / dt Rate of energy being used. Units: P = E t Watts = Joules/seconds If a differential amount of charge dq is given a differential increase in energy dw, the potential of the charge is increased by: By definition of current: I = dq / dt dw dq dw / dt = = P = V I dq dt w = t Pdt total energy V = dw / dq A very practical formulation! If we would like to know total energy 6 Basics Warning! In everyday language, the term power is used incorrectly in place of energy. Power is not energy. Power is not something you can run out of. Power can not be lost or used up. It is not a thing, it is merely a rate. It can not be put into a battery any more than velocity can be put in the gas tank of a car. Metrics How do we measure and compare power consumption? One popular metric for microprocessors is: MIPS/watt MIPS, millions of instructions per second.» Typical modern value? Watt, standard unit of power consumption.» Typical value for modern processor? MIPS/watt is reflective of the tradeoff between performance and power. Increasing performance requires increasing power. Problem with MIPS/watt» MIPS/watt values are typically not independent of MIPS techniques eist to achieve very high MIPS/watt values, but at very low absolute MIPS (used in watches)» Metric only relevant for comparing processors with a similar performance. One solution, MIPS /watt. Puts more weight on performance. 7 8 Metrics How does MIPS/watt relate to energy? Average power consumption = energy / time MIPS/watt = instructions/sec / joules/sec = instructions/joule therefore an equivalent metric (reciprocal) is energy per operation (E/op) E/op is more general - applies to more than processors also, usually more relevant, as batteries life is limited by total energy draw. This metric gives us a measure to use to compare two alternative implementations of a particular function. 9 Power in CMOS Switching Energy: energy used to switch a node Calculate energy dissipated in pullup: E sw = = cv P( t) dt = dv c ( V Energy supplied V pullup network pulldown network GN 0 C i(t) v(t) v) i( t) dt = v dv = cv V v(t) ( V cv Energy stored = cv v) c ( dv dt) dt = Energy dissipated An equal amount of energy is dissipated on pulldown. 0 5

6 Switching Power Gate power consumption: Assume a gate output is switching its output at a rate of: P /f = E t = switching rate E sw Therefore: P = α f cv P Chip/circuit power consumption: P = n α f cv number of nodes (or gates) α f activity factor clock rate (probability of switching on any particular clock period) clock f Vin Other Sources of Energy Consumption Short Circuit Current: I Vout Vout Vin Vin 0-0% of total chip power Junction iode Leakage: ~nwatt/gate few mwatts/chip I Vin=0 Transistor drain regions leak charge to substrate. evice Ids Leakage: Ioff Vout=V I iode Characteristic Ids V Vth Transistor s/d conductance never turns off all the way. ~pwatts/transistor. ~mwatt/chip Low voltage processes much worse. Vgs Controlling Energy Consumption What control do you have as a designer? Largest contributing component to CMOS power consumption is switching power: P = n α Factors influencing power consumption: n: total number of nodes in circuit α: activity factor (probability of each node switching) f: clock frequency (does this effect energy consumption?) V : power supply voltage f cv What control do you have over each factor? How does each effect the total Energy? Power / Cost / Performance Parallelism to trade cost for performance. As we trade cost for performance what happens to energy? 0. m 0. mt 0. mt 0. proj grade m mt 0. proj mt acc = m mt; acc = acc mt; 0. acc = 0. acc; acc = 0. proj; grade = acc acc; E MUL E A E WIRES E MUL E A E WIRES E MUL E A E MUXES E CNTL E WIRES The lowest energy consumer is the solution that minimizes cost without time multipleing operations. grade controller m m mt proj ALU acc acc 6

EECS Components and Design Techniques for Digital Systems. Lec 26 CRCs, LFSRs (and a little power)

EECS Components and Design Techniques for Digital Systems. Lec 26 CRCs, LFSRs (and a little power) EECS 150 - Components and esign Techniques for igital Systems Lec 26 CRCs, LFSRs (and a little power) avid Culler Electrical Engineering and Computer Sciences University of California, Berkeley http://www.eecs.berkeley.edu/~culler