Number systems
- Place values
- binary 2k, octal 8k, hex 16k
- To decimal
- Σ dk·bk
- From decimal
- divide by the base, read the remainders bottom-up
- Hex ↔ binary
- one hex digit = 4 bits
- Two’s complement
- invert the bits and add 1
Digital Electronics: cram pack
Formula sheets, the key method steps for every topic and 6 practice problems, with full solutions at the back.
1Formulas and methods
Digital logic: gates, K-maps & flip-flops
- Basic gates
- A·B, A + B, A′, (A·B)′, (A + B)′, A ⊕ B
- Boolean rules
- X + X′ = 1, X·X′ = 0, X + XY = X, XY + XY′ = X
- De Morgan
- (X·Y)′ = X′ + Y′, (X + Y)′ = X′·Y′
- Karnaugh map order
- 00, 01, 11, 10 (Gray code)
- Group of 2ᵏ cells
- loses k variables
- 4-to-1 multiplexer
- F = A′B′I0 + A′BI1 + AB′I2 + ABI3
- Flip-flops
- D: Q⁺ = D, T: Q⁺ = T ⊕ Q, JK: Q⁺ = JQ′ + K′Q
- Counters
- n-bit: 2ⁿ states; mod-N: smallest n with 2ⁿ ≥ N; fout = fclk/N
- Ripple counter
- fmax = 1/(n·tpd)
- Ring and Johnson counters
- n and 2n states
Method
- For a circuit, work from the inputs to the output one gate at a time and write the value on every wire, or write its expression and fill in the truth table.
- For simplification, put the 1s (and the don’t-cares as X) into a Karnaugh map with Gray-code rows and columns: 00, 01, 11, 10.
- Cover the 1s with as few and as large groups of 1, 2, 4, 8 or 16 cells as possible, wrapping round the edges; each group is one product term made of the variables that stay constant. Group the 0s for the POS form.
- For flip-flops and counters, read the inputs at each active clock edge, write the state after every edge, and remember that an n-bit counter wraps round after 2ⁿ states.
Data acquisition & sampling
- Nyquist criterion
- fs > 2·fmax, fN = fs/2
- Alias frequency
- fa = |f − k·fs|, k = round(f/fs)
- Step size
- 1 LSB = VFSR/2ᴺ
- Quantisation error (rounding)
- ±½ LSB
- Truncating ADC code
- D = ⌊Vin/LSB⌋
- Bits for a resolution r
- 2ᴺ ≥ span/r
- Ideal SQNR, full-scale sine
- SQNR = 6.02·N + 1.76 dB
- Data rate
- channels × fs × N bit/s
- Multiplexed ADC
- fs = 1/(n·(tc + tset))
- UART
- Tb = 1/baud; 8N1 = 10 bits per byte; bytes/s = baud/bits per frame
- Baud rate divider
- baud = fclk/(16·D)
Method
- Write down what is given: sampling rate, signal frequencies, number of bits, input range, baud rate.
- For sampling questions, compare each frequency with fs/2; a higher frequency appears at fa = |f − k·fs| with the nearest multiple of fs.
- For resolution questions, find 1 LSB = VFSR/2ᴺ, or the smallest N with 2ᴺ ≥ span/resolution.
- For data rates and UART links, count everything that is sent: channels, samples per second, bits or whole bytes per sample, and the start, parity and stop bits of every frame.
2Practice problems
1.Number systemsMedium
Set the bits so they store −14 in 8-bit two's complement. Read as an unsigned number, what is the pattern's value U?
N = −148 bit
U =
2.Number systemsHard
An uncompressed image is 1024 × 768 pixels with 16 bits per pixel. How much memory M does it take, in KiB (1 KiB = 1024 bytes)?
1024 × 768 px16 bit/px1 KiB = 1024 B
M = KiB
3.Digital logic: gates, K-maps & flip-flopsMedium
In this Karnaugh map exactly one 1 has no neighbouring 1, so it stays a group of one cell (a full minterm of four literals) in the minimal SOP form. Neighbours across the map edges count. Click that cell.
F = Σm(0, 3, 5, 6, 8, 10, 11, 13, 15)
- 1AB = 00, CD = 00
- 2AB = 00, CD = 01
- 3AB = 00, CD = 11
- 4AB = 00, CD = 10
- 5AB = 01, CD = 00
- 6AB = 01, CD = 01
- 7AB = 01, CD = 11
- 8AB = 01, CD = 10
- 9AB = 11, CD = 00
- 10AB = 11, CD = 01
- 11AB = 11, CD = 11
- 12AB = 11, CD = 10
- 13AB = 10, CD = 00
- 14AB = 10, CD = 01
- 15AB = 10, CD = 11
- 16AB = 10, CD = 10
4.Digital logic: gates, K-maps & flip-flopsHard
Find the minimal SOP form of F(A, B, C, D) = Σm(0, 2, 4, 6, 9, 14, 15) + d(1, 12), where d marks the don't-care conditions. How many literals L does it contain?
F(A, B, C, D) = Σm(0, 2, 4, 6, 9, 14, 15) + d(1, 12)
L =
5.Data acquisition & samplingMedium
Find the ideal signal-to-quantisation-noise ratio SQNR of a 16-bit ADC for a full-scale sine wave.
N = 16
SQNR = dB
6.Data acquisition & samplingHard
A microcontroller with a 16 MHz clock sets its UART baud rate as fclk/(16·D), with D a whole number rounded to the nearest value. The target is 38400 baud. Find the percentage error of the real baud rate (positive when it is too fast).
fclk = 16 MHztarget = 38400 Bd
e = %
3Solutions
1.Number systems
Answer: U = 242
Setting bits: each bit is worth a power of two (128, 64, …, 1). One hex digit is exactly four bits, and in 8-bit two's complement the leftmost bit is worth −128. Bitwise operations work on each bit position separately.
Invert and add one
14 = 0000 1110
inverted: 1111 0001
+1 → 1111 0010
Read it as unsigned
1111 0010₂ = 256 − 14 = 242
Answer
U = 242
2.Number systems
Answer: M = 1536 KiB
Positional number systems: each digit is worth its value times a power of the base. In binary the places are worth 1, 2, 4, 8, 16, …; in hexadecimal 1, 16, 256, …, with A–F standing for 10–15. In 8-bit two's complement the leftmost bit is worth −128 instead of +128, which is how negative numbers are stored.
Bits and bytes
1024·768·16 = 12582912 bit = 1572864 B
Convert to KiB
1572864/1024 = 1536 KiB
Answer
M = 1536 KiB
3.Digital logic: gates, K-maps & flip-flops
Answer: AB = 01, CD = 10
Karnaugh maps: the cells are ordered in Gray code (00, 01, 11, 10), so neighbouring cells, including the ones across the edges, differ in exactly one variable. Group the 1s in rectangles of 1, 2, 4, 8 or 16 cells, as large as possible and as few as possible, until every 1 is covered. Each group is one product term: the variables that stay constant inside it. A group of 2ᵏ cells loses k variables. Don't-cares (X) may be used to make groups bigger but never have to be covered. Grouping the 0s instead gives the minimal POS form.
Neighbours of each 1
A 1 with a neighbouring 1 can join a group of two or more.
m6: m7, m4, m2, m14 → 0
The isolated 1
m6 = A′BCD′ → AB = 01, CD = 10
Answer
AB = 01, CD = 10
4.Digital logic: gates, K-maps & flip-flops
Answer: L = 8
Karnaugh maps: the cells are ordered in Gray code (00, 01, 11, 10), so neighbouring cells, including the ones across the edges, differ in exactly one variable. Group the 1s in rectangles of 1, 2, 4, 8 or 16 cells, as large as possible and as few as possible, until every 1 is covered. Each group is one product term: the variables that stay constant inside it. A group of 2ᵏ cells loses k variables. Don't-cares (X) may be used to make groups bigger but never have to be covered. Grouping the 0s instead gives the minimal POS form.
Prime implicants
The largest groups that can be formed from 1s and Xs.
BD′, B′C′D, A′D′, A′B′C′, ABC
Minimal cover
F = B′C′D + A′D′ + ABC
L = 8
Answer
L = 8
5.Data acquisition & sampling
Answer: SQNR = 98.08 dB
ADC resolution: an N-bit converter divides its full-scale range VFSR into 2ᴺ codes, from 0 to 2ᴺ − 1. One step is 1 LSB = VFSR/2ᴺ: the smallest change the converter can show. A truncating ADC gives the code ⌊Vin/LSB⌋; a rounding ADC is never more than ½ LSB away from the true value, its quantisation error. Every extra bit halves the LSB and adds about 6 dB to the ideal signal-to-quantisation-noise ratio SQNR = 6.02·N + 1.76 dB for a full-scale sine. To reach a resolution r over a span, choose the smallest N with 2ᴺ ≥ span/r.
Ideal SQNR
SQNR = 6.02·16 + 1.76 = 98.08 dB
Answer
SQNR = 98.08 dB
6.Data acquisition & sampling
Answer: e = 0.1603 %
UART: an asynchronous serial link sends each byte as a frame: one start bit (low), the data bits, an optional parity bit and one or two stop bits (high). The baud rate is the number of bits per second, so one bit lasts Tb = 1/baud and a frame of n bits lasts n/baud. The common 8N1 format has 10 bits per byte. The baud generator divides the clock: baud = fclk/(16·D) with a whole number D, so the real baud rate differs slightly from the target; an error below about 2 % is usually acceptable.
Divider
D = 16·10⁶/(16·38400) = 26.04 → 26
Real baud rate
16·10⁶/(16·26) = 38462 Bd
Error
(38462 − 38400)/38400·100 = 0.1603 %
Answer
e = 0.1603 %
An error below about 2 % is usually acceptable.
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