1. Shift registers and Floating Point Numbers
Chapter 11 in Tokheim
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2. What are they?
Recall that a register is a small piece of memory that holds values.
In addition to holding values, a shift register performs a simple operation on the values; it moves them to the left or to the right.
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3. Example
Serial output
Serial input
Shift register 1
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4. Parallel loading register
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5. Register
On the previous slide, the input of a flip-flop is selected from two possible choices
The output of the same flip-flop.
The data switch above.
Recall that selecting from one of two inputs is done by a 2-to-1 MUX.
The load line serves as the address/select input.
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6. When Load is high, we are selecting the data switches to be the data input. So when we go through a positive edge of the clock, we are writing the value from the data switches to the register.
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7. When Load is low, we are selecting the flip-flop’s output to be the data input. So when we go through a positive clock edge, we are writing the value from the flip-flop to the flip-flop – thus keeping the value the same as before (holding).
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8. Register  Shift Register
We can adapt the previous circuit to make a shift register.
Instead of having one of the two possible inputs for a flip-flop come from the output of the same flip-flop, we can change this to having that input come from an adjacent flip-flop.
Then if when the load input is low and we go through a positive clock edge, the effect is not to hold the values of the register but to shift them.
This is part of the lab.
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9. On the ends
If one shifts from the right to the left, then the input to the rightmost flip-flop does not come from an adjacent flip-flop during the shift operation. There are several options
Data switch input
Always 1
Always 0
Use leftmost output to form a ring
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10. How are shift registers used?
Modems
Cyclic Redundancy Check (CRC)
Multiplication
Adding floats
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11. Modems
A modem (Modulator-Demodulator) takes a signal from a computer and places it on a transmission line.
A transmitting modem modulates, that is, converts a digital signal from a computer to a pseudo-analog signal more appropriate for a transmission line.
The receiving modem demodulates, that is, converts the pseudo-analog signal back into digital form.
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12. Modems (Cont.) But the aspect of modems relevant here is that
The transmitting modem converts parallel data to serial.
The receiving modem converts serial data into parallel form.
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13. Inside  Outside
Inside the computer, data that moves around as words on parallel cables having a connection for each bit in the word.
(Although with SATA, some internal data movement has become serial.)
The transmission lines are longer and require data to be sent serially (one bit at a time).
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14. Parallel to Serial
To leave the computer, data moves into register in parallel, several bits at once.
Data then moves out of the register serially, one bit at a time. 1
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15. Serial to Parallel 1 DEMO!
To enter the computer, data enters the register in serial, one bit at a time.
Data then moves out of the register in parallel.
DEMO!
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16. Help for 74165
S/L should be low to get the data into the 74165, then S/L should be high for the shifting action to occur. The loading or shifting occurs on the edge of the clock.
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17. Help for 74164
If CLR is high, and both A and B are high, a shifting occurs with an high coming in as the new bit. If either A or B is low shifting occurs with a low coming in as the new bit. That’s why the “transmission line” is fed to both A and B inputs.
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18. “Modem” simulation
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19. Cyclic Redundancy Check
In order to check for errors that may occur during transmission, the sender calculates a number, a cyclic redundancy check. The receiver does the same calculation. If they agree, then presumably no error occurred in transmission.
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20. CRC Modification
Actually the receiver does a calculation that includes the sender’s CRC as part of the data and should get an answer of zero.
It’s easier electronically to see if series of bits corresponds to zero.
Any mathematical operation performed on the transmitted data could serve as a check.
Another common calculation is summing, then it is called a checksum.
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21. CRC: funny division The calculation should not be time consuming.
Think of CRC as a funny kind of division, the remainder from the division is the check.
It’s not ordinary division, but a strange kind of division that is easy to realize electronically.
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22. CRC = Shift register + XORs
Basically one has a shift register with a few excluded OR gates inserted in strategic positions.
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23.  
The first four shift in without anything interesting happening.   1 1   1
001010   1
01010
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24.   1
1010   1 1
010   1 1 1 10   1 1 1   1 1
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25. Multiplication: Shift and add1  +
shift
shift
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26. Fractions Similar to what we’re used to with decimal numbers 3.14159 =
3 · · · · · · 10-5 =
1 · · · · 2-2
+ 1 · · · · 2-6
(  )
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27. Converting decimal to binary II
98.61
Integer part
98 / 2 = 49 remainder 0
49 / 2 = 24 remainder 1
24 / 2 = 12 remainder 0
12 / 2 = 6 remainder 0
6 / 2 = 3 remainder 0
3 / 2 = 1 remainder 1
1 / 2 = 0 remainder 1
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28. Converting decimal to binary III
98.61
Fractional part
0.61  2 = 1.22
0.22  2 = 0.44
0.44  2 = 0.88
0.88  2 = 1.76
0.76  2 = 1.52
0.52  2 = 1.04
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29. Converting decimal to binary IV
Put together the integral and fractional parts
98.61 
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30. Another Example (Whole number part)
Integer part
123 / 2 = 61 remainder 1
61 / 2 = 30 remainder 1
30 / 2 = 15 remainder 0
15 / 2 = remainder 1
7 / 2 = remainder 1
3 / 2 = remainder 1
1 / 2 = remainder 1
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31. Checking: Go to Programs/Accessories/Calculator
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32. Put the calculator in Scientific view
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33. Enter number, put into binary mode
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34. Another Example (fractional part)
Fractional part
0.456  2 = 0.912
0.912  2 = 1.824
0.824  2 = 1.648
0.648  2 = 1.296
0.296  2 = 0.592
0.592  2 = 1.184
0.184  2 = 0.368 … …
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35. Checking fractional part: Enter digits found in binary mode
Note that the leading zero does not display.
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36. Convert to decimal mode, then
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37. Edit copy – View Scientific – Edit Paste
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38. Divide by 2 raised to the number of digits (in this case 7, including leading zero)
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39. In most cases it will not be exact
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40. Other way around
Multiply fraction by 2 raised to the desired number of digits in the fractional part. For example
.456  27 =
Throw away the fractional part and represent the whole number
58 111010
But note that we specified 7 digits and the result above uses only 6. Therefore we need to put in the leading 0
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41. Fixed point
If one has a set number of bits reserved for representing the whole number part and another set number of bits reserved for representing the fractional part of a number, then one is said to be using fixed point representation.
The point dividing whole number from fraction has an unchanging place in the number.
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42. Limits of the fixed point approach
Suppose you use 4 bits for the whole number part and 4 bits for the fractional part (ignoring sign for now).
The largest number would be =
The smallest, non-zero number would be = .0625
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43. Floating point representation
Floating point representation allows one to represent a wider range of numbers using the same number of bits.
It is like scientific notation.
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44. Scientific notation
Used to represent very large and very small numbers
Ex. Avogadro’s number
  1023 particles 
Ex. Fundamental charge e
  C
 C
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45. Scientific notation: all of these are the same number
=  100
 10 =  101
 100 =  102
 103
 104
Rule: Shift the point to the left and increment the power of ten.
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46. Small numbers
 10-1
 10-2
 10-3
 10-4
 10-5
1.234  10-6
Rule: shift point to the right and decrement the power.
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47. Floating Point Rules
Starting with the fixed point binary representation, shift the point and increase the power (of 2 now that we’re in binary).
Shift so that the number has no whole number part and also so that the first fractional bit (the half’s place) has a 1.
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48. Floats
SHIFT expression so it is just under 1 and keep track of the number of shifts
 27
Express the number of shifts in binary 
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49. Mantissa and Exponent and Sign
(Significand) Mantissa
Exponent
The number may be negative, so there a bit (the sign bit) reserved to indicate whether the number is positive or negative
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50. Biasing
Actually the exponent is not represented as shown on the previous slide.
There were 8 bits used to represent the exponent on the previous slide, that means there are 256 numbers that could be represented.
Since the exponent could be negative (to represent numbers less than 1), we choose half of the range to be positive and half to be negative , i.e to 127.
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51. Biasing (Cont.)
In biasing, one does not use 2’s complement or a sign bit.
Instead one adds a bias (equal to the magnitude of the most negative number) to the exponents and represents the result of that addition.
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52. Biasing (Cont.) With 8 bits, the bias is 128.
We had to shift 7 times to the left, corresponding to an exponent of +7
We add that to the bias 128+7=135
That is the number we put in the exponent portion:
(In the IEEE 754 format for floats, you bias by one less (127) and reserve the exponents and for special purposes.)
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53. Big floats
Assume we use 8 bits, 4 for the mantissa and 4 for the exponent (neglecting sign). What is the largest float?
Mantissa: Exponent 1111
 27
=120
(Compare this to the largest fixed-point number using the same amount of space )
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54. Small floats
Assume we use 8 bits, 4 for the mantissa and 4 for the exponent (neglecting sign). What is the smallest float?
Mantissa: Exponent 0000
0.5  2-8 =
(Compare this to the smallest fixed-point number using the same amount of space .0625)
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55. One more fine point
As discussed so far, the mantissa (significand) always starts with a 1.
When storage was expensive, designers opted not to represent this bit, since it is always 1.
It had to be inserted for various operations on the number (adding, multiplying, etc.), but it did not have to be stored.
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56. Adding floats Image adding two scientific notation numbers
× ×10-2
You must adjust the numbers so the have the same power – and you always pick on the small guy
× ×102
×102
Hey that’s shifting
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57. Suppose you had seven places to play with
× ×102
×102
Which is different from ×102
Rounding occurred – that’s why you shift the small number – you want to lose less significant information
NEVER TEST EQUALITY OF FLOATS
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