Powers of 2

1. Powers of 2

2. 1010010100011011 A 1 in the binary number is that place value’s power of 2 in decimal:215 214 213 212 211 210 29 28 27 26 25 24 23 22 21 20 32,768 16,384 8192 4,096 2,048 1,024 512 256 128 64 32 16 8 4 2 1 1 0 1 0 0 1 0 1 0 0 0 1 1 0 1 1 Just add up the place values with 1’s: 32,768 + 8192 + 1,024 + 256 + 16 + 8 + 2 + 1 = 91,419 Binary to Decimal Conversion

3. 12,472 Keep subtracting the highest place value that will leave a positive remainder, until the remainder is zero. If you subtract a place value, put a 1 at that place value in the binary number. If you can’t subtract a place value, put a 0 at that place value in the binary number. The most common mistake is to forget a place value, especially on the right: | read down | v v213 = 8,192 Highest place value, subtract it: 12,472 – 8192 = 4280 results in 1212 = 4,096 Can be subtracted from remainder: 4,280 – 4096 = 184 results in 1 211 = 2,048 Remainder is too small to subtract. results in 0 210 = 1,024 Remainder is too small to subtract. results in 0 29 = 512 Remainder is too small to subtract. results in 0 28 = 256 Remainder is too small to subtract. results in 0 27 = 128 Can be subtracted from remainder: 184 – 128 = 56 results in 126 = 64 Remainder is too small to subtract. results in 0 25 = 32 Can be subtracted from remainder: 56 – 32 = 24 results in 124 = 16 Can be subtracted from remainder: 24 – 16 = 8 results in 1 23 = 8 Can be subtracted from remainder: 8 – 8 = 0 results in 1 22 = 4 Remainder is too small to subtract <= don’t forget these! results in 0 21 = 2 Remainder is too small to subtract <= don’t forget these! results in 0 20 = 1 Remainder is too small to subtract <= don’t forget these! results in 0 The answer is the binary digits, in order, with the highest place value to the left:1000010111000 Decimal to Binary Conversion

4. Digits Hex Dec Binary 0 = 0 = 0000 1 = 1 = 0001 2 = 2 = 0010 3 = 3 = 0011 4 = 4 = 0100 5 = 5 = 0101 6 = 6 = 0110 7 = 7 = 0111 8 = 8 = 1000 9 = 9 = 1001 A = 10 = 1010 B = 11 = 1011 C = 12 = 1100 D = 13 = 1101 E = 14 = 1110 F = 15 = 1111 Conversion from Binary to Hexadecimal (Hex to Binary conversion is just the opposite) : 111001010001101110101110100010 Split into groups of four, right to left: 11 1001 0100 0110 1110 1011 1010 0010 Put enough zeros on the far left to fill the left group: 0011 1001 0100 0110 1110 1011 1010 0010 Convert to Hexadecimal by replacing each group with the correct single hex digit: 3 9 4 6 E B A 2 Hexadecimal equivalent (represents 32 bit word): 3946 EBA2 Hexadecimal

5. Metric Prefixes (whole numbers)

6. Metric Prefixes (fractions)

7. y = log(x) with x and y axes equal linear scale

8. y = log(x) with x and y axes linear scaley axis expanded

9. y = log(x) with x axis logarithmic scale and y axis linear

10. Bels and deciBels (positive)

11. Bels and deciBels (negative)

12. dB references • dBW – dB Watts (0dBW = 1 Watt) • dBm – dB milliWatts (0dBm = 1/1000 Watt) • dBi – antenna gain (0dBi = the field intensity of an isotropic antenna at a given input power and frequency) • An isotropic antenna is a simple straight wire antenna without any physical modifications. • A differently shaped antenna’s field intensity is measured and compared to the isotropic antenna at the same input power and frequency. The ratio is expressed in deciBels as dBi.