Chapter 5: Signal conversion

Transcription

1 Chapter 5: Signal conversion Learning Objectives: At the end of this topic you will be able to: explain the need for signal conversion between analogue and digital form in communications and microprocessors describe how an op-amp summing amplifier can be used as a digital-to-analogue converter (DAC) to convert a digital signal into an analogue signal analyse and design a DAC based upon an op-amp summing amplifier to meet a given specification describe how comparators can be used as an ADC to convert an analogue signal into a digital signal describe the process of digitising audio signals and explain the effects of sampling rate and resolution analyse and design a flash converter ADC based on comparators and priority encoders to meet a specification and describe the factors affecting the resolution select and apply the equation for calculating the resolution of a n-bit flash converter: i/p voltage range resolution = n 2 compare the difference of a digital ramp ADC and a flash ADC. 170

2 Digital and analogue information: The differences between analogue and digital signals were described earlier, in the AS E-book. In most situations, real world information exists in analogue form, not digital. Sound signals, for example, such as speech and music, are analogue. However, they are often converted into digital format - before being transmitted through a communication system or being stored for example. At the end of the process, the digital signals can be converted back to recover the original analogue signal. The advantage in doing so is that, apart from minor losses inherent in the conversion process, no other loss of quality occurs. On the other hand, an analogue signal suffers some loss of quality each time it is copied or processed. The conversion process involves two sub-systems - an analogue-to-digital converter (ADC), which converts information from analogue into digital format and a digital-to-analogue converter (DAC), which converts a digital signal to an analogue signal. A block diagram of a typical audio system is shown below: Control systems have a different function. They use sensors, monitoring quantities such as temperature, light intensity, pressure or humidity, to control output devices such as motors and lamps. To do so: the analogue signals from the sensor are converted to digital format these are processed digitally they are then converted back to analogue format for the output signal. In some cases, the resulting output signal simply switches a device on or off. Sometimes, it delivers variable control, making a motor rotate faster or slower, making a piezo sound louder or quieter or a lamp brighter or darker. A block diagram of a typical control system is shown below: Analogue sensing units: In the E-book for the AS course, the behaviour of sensing units using thermistors, LDR s and phototransistors was examined. 171

3 Analogue-to-digital conversion: The ADC is a system which outputs a bigger and bigger binary number as the analogue input voltage gets bigger and bigger. Usually, the binary number produced, doubles when the input voltage doubles i.e. when the input changes from V IN to 2V IN, the output changes from nnnn to 2 x (nnnn). There are a number of ways to design such a circuit. One type, the flash converter is based on the behaviour of comparators, studied earlier. Comparators These devices were described earlier in the AS course. The output of a comparator is always saturated, either in positive saturation - as close as possible to the positive supply voltage - or in negative saturation - as close as possible to the negative supply voltage. The relative size of the two input voltages decides which of these occur. Often, the inverting input is provided as a reference voltage, V ref. The input voltage, V IN, connected to the non-inverting input is compared with this reference voltage. With this arrangement: V OUT = +V SAT when V IN is bigger than V ref. VOUT = -VSAT when V IN is smaller than V ref. where: +V SAT = positive saturation voltage -V SAT = negative saturation voltage. Op-amps obey the following relationship: V OUT = G 0 (V IN V ref ) where G 0 is the open loop gain of the op-amp (usually > 10 6 ). Using this equation when the op-amp has an output saturation voltage of 12V, the minimum difference between V IN and V ref that changes V OUT from -V SAT to +V SAT is: V (V IN V ref ) = OUT G 0 = = 12 mv The comparator can be thought of as 1-bit ADC and is the basic building block of a flash converter. 172

4 Adding more comparators The following circuit shows a flash converter consisting of three comparators connected to a resistor chain of four resistors. The resistors form a voltage divider chain, which applies a different voltage, V REF, to each comparator. This chain is connected to a reference voltage V REF. In the circuit diagram, V ref1 = 0.5 V, V ref2 =1.0 V and V ref3 = 1.5 V. The range of analogue input voltages should not exceed V REF. The graphs show how the comparator outputs change as the analogue input, V IN, is increased steadily from 0 V to 2 V: Although the three outputs are digital they do not form a binary sequence. 173

5 Assuming that a voltage of +V SAT corresponds to logic 1 and -V SAT to logic 0, the following table shows the sequence created: Analogue input voltage, V IN Output A Output B Output C 0 < V IN < < V IN < < V IN < V IN > To design a flash converter that produces a binary output, an extra block called a priority encoder is needed. 2-bit Flash Converter The basic circuit diagram for a 2-bit flash converter is shown below: Notice: the analogue input signal is connected to the non-inverting inputs of the comparators their inverting inputs are connected to various points on a resistor chain, which creates a different reference voltage for each comparator. Each comparator compares the analogue input signal voltage, V IN, with its reference voltage, and outputs a voltage at either positive or negative saturation level, following the rules outlined above. At this stage, the analogue signal has been converted into a set of digital signals, but not into a pure binary number. The outputs of the comparators become inputs for the priority encoder, a combinational logic system which converts the comparator outputs into a binary number. The uppermost comparator in the system is used to indicate that the analogue signal has exceeded the voltage range that the ADC can handle. Usually, it does so by lighting an overflow LED. 174

6 3-bit Flash Converter As the number of output bits increase, the number of comparators and the complexity of the priority encoder increase significantly. The circuit diagram for a 3-bit flash converter is shown below: The main advantage of the flash converter is its speed. The conversion time, to generate the binary number from an analogue signal, is very short, typically 100ns or less. The analogue signal is connected directly to each comparator. Their outputs switch at the same time into either positive or negative saturation. (For this reason, this type of ADC is also known as a parallel A to D converter.) The priority encoder consists of a number of logic gates, or equivalent, which rapidly generate the required binary number at the output. The main disadvantage is its cost. This is the result of the complexity of the circuit. The three-bit ADC, shown above, can output eight different binary numbers, from 000 to 111. Each of these corresponds to a small range of analogue input voltages. For example, if the three-bit ADC is designed for analogue signals in the range 0 to 8 V, then 000 would correspond to voltages from 0 V to 1 V, 001 from 1 V to 2 V, and so on. The circuit must be able to distinguish between these voltage ranges. To do so, a comparator is set up to respond to each of these ranges. This needs seven comparators, plus one more to detect the overflow condition. In general, a n-bit ADC requires (2 n 1) comparators connected to the priority encoder, plus one for the overflow detection, making a total of 2 n comparators. Most digital systems manipulate data in the form of eight-bit binary numbers, or more. For an eight-bit flash converter, 2 8 (= 256) comparators are needed. Hence it is an expensive circuit. This type of ADC is widely used in applications such as digitising music or video signals in TV tuner cards, for example, where high conversion speed is essential. 175

7 Analysing an ADC circuit: Here is the same 3-bit ADC circuit diagram, but with component values, reference voltage, and labels, added. The voltage dropped across the resistor chain V REF = 0.8 V. The voltage dropped across each resistor = 0.8/8 = 0.1 V. Hence: voltage at A = 0.1V voltage at B = 0.1 X 2 = 0.2 V voltage at C = 0.1 X 3 = 0.3 V voltage at D = 0.1 X 4 = 0. 4V. Assuming that the positive saturation voltage is +12 V, and negative saturation voltage is 0 V, the behaviour of the comparators is described in the table below: Analogue input V IN Comparator outputs / V S T U V W X Y V IN < 0.1V V < V IN < 0.2V V < V IN < 0.3V V < V IN < 0.4V V < V IN < 0.5V V < V IN < 0.6V V < V IN < 0.7V V < V IN < 0.8V V IN > 0.8V Z =+12V indicating overflow Note: The maximum analogue input voltage that can be applied to an ADC without causing an overflow is equal to V REF. 176

8 The priority encoder: The priority encoder is a combinational logic system, which inputs signals from the comparator outputs and processes them into binary numbers. Assuming that positive saturation (+12 V) corresponds to logic 1 and negative saturation, (0V) to logic 0, the priority encoder for the three-bit ADC shown earlier, obeys the following truth table: Analogue input V IN Priority encoder inputs (= comparator outputs) Priority encoder output S T U V W X Y C B A V IN < 0.1V V < V IN < 0.2V V < V IN < 0.3V V < V IN < 0.4V V < V IN < 0.5V V < V IN < 0.6V V < V IN < 0.7V V < V IN < 0.8V We do not need to consider output Z, from the overflow comparator, as it is not connected to the priority encoder and the Analogue input column is shown only for reference as it plays no part in the design. Converting the truth table into a combinational logic system is potentially a very complex problem. With seven inputs, there are theoretically 2 7 (= 128) possible input combinations. In reality, only the eight shown in the truth table can ever occur. The remaining 120 combinations can never occur and so are don t care states, when it comes to generating Boolean expressions for the logic system. For example, when the analogue voltage, V IN, is greater than 0.3V (making U change to positive saturation,) it is also greater than 0.2V (turning T to positive saturation,) and greater than 0.1V (turning S to positive saturation.) In other words, all rows where U = 1 but S and T are logic 0 cannot exist. By inspection, the following logic expressions are obtained: C = V B = T.V + X A = S.T + U.V + W.X + Y 177

9 The outputs C, B and A could be generated using appropriate logic gates, or using a multiplexer or memory IC. The logic gate solution for output A is shown below: ADC Resolution: This is a measure of the sensitivity of the ADC, i.e. a measure of the minimum change in the input signal that will guarantee a change in the output signal. It is closely connected to two other parameters: the number of bits in the output the analogue input voltage range for the ADC (determined by V REF ). Consider a three-bit ADC with an input voltage range of 4.0V. It has eight possible outputs 000, 001, 010, 011, 100, 101, 110 and 111. As the analogue input voltage changes: inputs smaller than 0.5 V produce an output number of 000 inputs of 0.5 V, or up to 1.0 V, cause the output to be 001 inputs of 1.0 V, or up to 1.5 V, cause the output to be 010. and so on until: inputs of 3.0 V, or up to 3.5 V, cause the output to be 110 inputs greater than 3.5 V cause the output to be 111. When the input voltage exceeds 4.0 V, the output will still be 111 but the overflow indicator will be activated. Here the resolution is 0.5 V. Whenever the input changes by more than 0.5 V, there must be a change in the output number. A change of less than that may or may not cause the output to change. (Suppose that the input sits at 0.9 V, and then increases by 0.2 V. The output will change from 001 to 010. However, if the input sits at 1.2 V and increases by 0.2 V, the output will not change.) 178

10 Conversion of a continuously varying analogue signal into a discrete number of voltage levels is referred to as quantization. It creates an error called the quantization error, which ranges from zero to a maximum equal to the resolution. Average quantization error = resolution 2 For a n-bit ADC, of the type considered so far: Resolution = input voltage range 2 n For example, for an eight-bit flash converter of this type with an input voltage range of 10 V, Resolution = i/p voltage range n 8 Average quantization error = 20 mv The smaller the resolution the smaller the quantization error = = = V = 39 mv Examples 1. A five-bit flash converter has an input voltage range of 2.0 V. What is its resolution? Resolution = i/p voltage range 20. = = V n What is the input voltage range of an eight-bit flash converter having a resolution of 0.05 V? i/p voltage range Resolution = n 2 n Input voltage range = resolution x 2 8 = x 2 = V 3. An ADC is required to have a minimum resolution of 1 mv. The input voltage range is 3 V. What is the minimum number of bits needed for the ADC to provide this resolution? Resolution = n 2 = i/p voltage range n 2 i/p voltage range n 2 = 1 mv n 2 = 3000 resolution 3 V 179

11 Method 1: Using logarithms Taking logs of both sides: n x log 10 2 = log Therefore, choose a 12-bit ADC n = log log 10 2 = 11.6 Method 2: Trial and error: 2 n = 3000 Try n = 9: 2 9 = 512 too small Try n = = 1024 too small Try n = = 2048 too small Try n = = 4024 greater than the minimum required Therefore, choose a 12-bit ADC. 180

12 Investigation 5.1 Set up the following ADC circuit on Circuit Wizard. (b) Adjust V1 to 0.5 V. Measure the voltages at X, Y and Z and the logic levels at A, B and the Overflow. Record the results in the table. Analogue input V I V1 = 0.5 V V1 = 1.5 V V1 = 2.5 V V1 = 3.5 V V1 = 4.5 V Comparator output voltage Binary output X Y Z B A Overflow (c) Complete the table for the other values of V1. (d) Comment on the performance of the ADC. (e) What is the resolution of the ADC? 181

13 Designing a flash ADC to meet a given specification: The specification for an ADC includes: the operational input voltage range the number of bits in the output binary number the resolution the speed of conversion the power supply used. Here, we design an ADC with the following parameters: number of output bits = 4 input voltage range = 0 to 4 V comparator outputs: positive saturation = +10 V negative saturation = 0 V A four-bit flash converter requires sixteen (=2 4 ) comparators, and hence sixteen equal sized resistors in the resistor chain, (for a linear response - output number directly proportional to the input voltage.) The input voltage range dictates the size of the reference voltage. Hence, the reference voltage is 4 V. The resolution is 4 = 0.25 V, using the formula given earlier. 16 The sixteen resistors divide the 4 V reference voltage into 0.25 V portions. The actual value of resistor used in the chain is not important. Too small a value would mean that large currents flow, dissipating large amounts of power. As a rule of thumb, the resistors should always be at least 1 kω. In this case, 10 kω resistors are used. The circuit diagram for the ADC is shown opposite, though not drawn in full. 182

14 With fifteen inputs, there are theoretically 2 15 (= 32768) possible input combinations. Thankfully only the sixteen shown in the table that follows can ever occur. Analogue input V IN Priority encoder inputs (= comparator outputs) Priority encoder output K L M N O P Q R S T U V W X Y D C B A V IN < 0.25 V V< V IN < 0.50 V V< V IN < 0.75 V V< V IN < 1.00 V V< V IN < 1.25 V V< V IN < 1.50 V V< V IN < 1.75 V V< V IN < 2.00 V V< V IN < 2.25 V V< V IN < 2.50 V V< V IN < 2.75 V V< V IN < 3.00 V V< V IN < 3.25 V V< V IN < 3.50 V V< V IN < 3.75 V V< V IN < 4.00 V V IN > 4.00 V Z = +10 V indicating overflow The priority encoder is designed in the same way as before, by examining the truth table. This time, we assume that a comparator output voltage of +10 V is seen as logic 1, and an output of 0 V is seen as logic 0 by the priority encoder. Here are logic expressions for the priority encoder: D = R C = N.R + V B = L.N + P.R + T.V + X A = K.L + M.N + O.P + Q.R + S.T + U.V + W.X + Y The overflow indicator functions as in the previous example. (The logic expressions for the 4-bit priority encoder are provided for illustration purposes only. You are not expected to be able to design a 4-bit priority encoder.) 183

15 Exercise (a) A five-bit flash converter has an input voltage range of 1.5 V. What is its resolution? (b) What is the input voltage range of a ten-bit flash converter that has a resolution of 0.04 V? (c) An ADC is required with a minimum resolution of 500 mv. The input voltage range is 2.0 V. What is the minimum number of bits needed for this ADC? 2. The diagram shows part of the circuit for a 2-bit flash ADC. The system meets the following specification: Analogue Voltage Voltage Voltage Binary output input V IN at L at M at N B A 0 to 0.25 V 0 V 0 V 0 V V to 0.50 V 0 V 0 V 12 V V to 0.75 V 0 V 12 V 12 V V to 1.00 V 12 V 12 V 12 V

16 (a) Complete the circuit diagram by: adding a fourth comparator so that its output indicates an overflow condition, when the analogue input voltage exceeds 1.0 V adding all connections needed labelling the inverting inputs of the op-amps with a - and the non-inverting inputs with a +. (b) Calculate suitable values for resistors P, Q, R and S. Resistor P = Resistor Q = Resistor R = Resistor S = 3. Here is the circuit diagram for an Analogue-to-Digital converter (ADC). (a) What is the function of the following components in this system: (i) the priority encoder (ii) the comparator whose output is labelled W? (b) Calculate the resolution of this ADC. (c) What is the analogue input voltage range for this ADC? 185

17 (d) When V REF is reduced to +1.0 V, what is the effect of this change on: (i) resolution? (ii) analogue input voltage range? (e) The comparators have saturation voltages of +6 V and +0 V and V REF is set at +1.0 V. The top graph shows how the input voltage V IN changes over a period of time. Use the axes provided to draw the signals at X, Y and Z over this time. 186

18 4. Design an ADC with the following parameters: number of output bits = 2 input voltage range = 0 to 3 V comparator outputs: positive saturation = +12 V negative saturation = 0 V Your design should include: a circuit diagram labelled with component values and reference voltage the resolution of the ADC a table showing the operation of the circuit Boolean expressions for outputs A and B of the priority encoder, expressed in terms of the inputs (Hint: Use a Karnaugh map and make use of the don t care states) the logic circuit for the priority encoder. Circuit diagram: 187

19 Circuit operation: Analogue input V IN V IN < V IN > < V IN < < V IN < < V IN < Comparator outputs /V Binary number at output B A Truth table for priority encoder: Comparator outputs Binary number at output B A Boolean expressions: Circuit diagram for priority encoder: 188

20 Flash ADC vs digital ramp ADC The principle of the digital ramp converter is illustrated in the following block diagram. When the Start conversion switch is pressed, the counter is reset and begins to count from zero. The DAC converts the binary number produced into an analogue voltage, which is compared to the analogue input voltage. The count continues until the output of the DAC is bigger than the analogue input voltage. The display shows the current binary number, a digitised measure of the analogue input voltage. The structure or operation of the digital ramp ADC will not be examined, but you need to be able to compare its performance with that of the flash ADC: for a flash ADC, conversion time is much shorter, typically a few microseconds, (determined by the time taken for a comparator to switch, plus the time for the logic gates in the priority encoder to process the input signals. For the digital ramp ADC, conversion time depends on the size of the analogue signal. As the counter starts from zero each time the bigger the analogue input voltage, the further it must count before the DAC output reaches that value) the flash ADC circuit is more complex because of the large number of comparators needed (for example, an eight-bit flash ADC requires 256 comparators) as a result of this, flash ADC s are more expensive. Digital-to-analogue converters A digital-to-analogue converter, (DAC), generates an analogue output voltage which increases as the binary number applied to the input increases. There are a number of ways of achieving this. The one outlined here uses a modified op-amp summing amplifier circuit. The op-amp summing amplifier: These devices were described earlier in the AS course. The diagram shows a two-input summing amplifier. The output voltage, V OUT, is calculated using the formula: V V OUT = -R 1 + V 2 F R 1 R 2 Further input signals can be added, each with its own input resistor, as shown in the circuit diagram opposite, for a four-channel summing amplifier. The formula for this circuit is: V V OUT = -R 1 + V 2 + V 3 + V 4 F R 1 R 2 R 3 R 4 189

21 Example: Calculate the output voltage, V OUT, for the following circuit: V By using V OUT = -R 1 + V 2 + V 3 + V 4 F R 1 R 2 R 3 R 4 the output voltage will be: Digital signals V OUT = = -7 V A digital signal is a two-state signal, consisting of a number of bits, each logic 0 (~ 0V) or logic 1 (~+V S ). The place value of the bit (how much it is worth,) depends on its position: the least significant bit (LSB) is worth either 0 (for logic 0) or 1 (for logic 1) the next, on the left is worth either 0 (for logic 0) or 2 (for logic 1) the next is worth 0 or 4 and so on and so forth. Example: Convert the binary number into the equivalent decimal number. 190

22 The DAC circuit takes into account these properties of digital signals: the input voltages take one of two values, either the value representing logic 0, usually 0 V, or that representing logic 1, usually close to the positive supply voltage (+V S ) the output voltage takes into account the place value of the logic 1 input signals, by having a voltage gain that reflects this place value in other words, if the input receiving the least significant bit (LSB) has a voltage gain of G, then the input connected to the next bit must have a gain of 2G, the next a gain of 4G and so on In DAC circuits based on the summing amplifier, this is achieved by successively reducing the size of the input resistor. When the LSB input resistor is R, the next input resistor will be R, the next R 2 4 and so on. The design is based on an inverting amplifier, and so, when using positive logic (logic 1 = +V S,) the output voltage is negative. it must be powered from a split power supply, offering voltage rails at +V S, 0V and V S to overcome the inversion, a second inverting amplifier often follows the first, often with a voltage gain of simply -1. The next diagram these ideas into a four-bit circuit shows incorporated DAC: Choosing resistor values for this circuit It is very easy to saturate the output in a circuit like this. The simplest way to avoid this is to use fractional voltage gains for the summing amplifier. In the above circuit, for example: input A, (LSB) has a voltage gain of 1 24 input B has a gain of 1 12 input C 1 6 input D (MSB)

23 Analysing a DAC circuit: To analyse the above circuit: think of it as four inverting amplifiers combined so that the output voltage is the sum of their outputs the second op-amp has a voltage gain of -1, and so simply reverses the polarity of the output signal assume that 0 V represents a logic 0 signal, and +12 V a logic 1. For example: 1. When the input is : A = 12 V and B = C = D = 0 V voltage gain on input A = - RF - 10 = R IN 240 so its output = - 12 x 10 = V 240 the other inputs are set to 0 V and so output 0 V. The final output = - ( ) = +0.5 V 2. When the input is : A = B = D = 12 V and C = 0 V voltage gain on input A = 10, and its output = - 12 x 10 = -0.5 V voltage gain on input B = 10, and its output = - 12 x 10 = -1.0 V voltage gain on input D = 10, and its output = - 12 x 10 = -4.0 V The final output = - (-0.5) + (-1.0) (-4.0) = +5.5 V 192

24 DAC output - summary The behaviour of the four-bit DAC can be summarised in two ways: as a table of output voltages Binary number input D C B A V 1 /V V OUT /V as a graph. Viewed as a graph, the results show the characteristic staircase waveform. The output is analogue - it gets bigger as the digital input number gets bigger. However, it does not change continuously but rises in steps, often referred to as quantization steps. The step size depends on the voltage gain of the least significant input of the DAC and the voltage that represents logic 1. R F Step size = V L1 R 1 V L1 = voltage corresponding to logic 1 R F = feedback resistance R 1 = input resistance for least significant input. Applying this formula to the example above, step size = 12 X 10k = 0.5 V - the same answer as that 240k obtained by calculating V OUT for an input of

25 Investigation 5.2 (a) Set up the 4-bit DAC circuit shown below. (b) Measure and record the readings on voltmeters V 1 and V OUT with all four switches open. Binary number input D C B A V 1 /V V OUT /V (c) Complete the table by closing the switches in the order shown and recording the readings on voltmeters V 1 and V OUT. (d) Comment on how well the results compare with the table given on page

26 Exercise Here is the circuit diagram for a three-bit DAC. Calculate the output voltage, V OUT, when the following binary numbers are applied to the input: (a) 011 (b) 101 (c) Here is the circuit diagram for a digital-to-analogue converter (DAC). The most significant bit of the binary number is applied to input C, and the least significant bit to input A. The outputs of the op-amps saturate at +12 V and -12 V. 195

27 (a) What is the gain of amplifier Y? (b) The following voltages are applied to inputs A, B and C: Calculate: V A = +5 V V B = 0 V V C = 0 V (i) V 1 (ii) V 2 (c) What is the maximum value of output voltage V 2 that this 3- bit DAC will produce? (d) The system uses +5 V to represent logic 1 and 0 V to represent logic 0. Use the axes provided to draw a graph showing the relationship between V 2 and the digital input signal, for the four values of input given. Indicate the scale you are using for the vertical axis. 196

28 Designing a DAC to meet a given specification: The performance of a DAC can be described in a number of ways, including: the number of bits that can be inputted the voltage range for the output the step size for the output voltage the speed of conversion the power supply used. Here, we design a DAC with the following parameters: number of input bits = 4 output voltage range = 0 to 12 V logic 1 = 8 V and logic 0 = 0 V power supply = +15 V / 0 V / -15 V. In general, a n -bit DAC will have 2 n output voltage levels, with 2 n 1 steps between them. As the graph given earlier shows, a four-bit DAC produces sixteen (= 2 4 ) output voltage levels, with fifteen steps (2 4 1) between them. These fifteen steps must cover the 0 to 12 V voltage range. Each step corresponds to a voltage change of (12 / 15) = 0.8 V. Using the formula quoted earlier: where so that: Step size = V L1 R F R 1 V L1 = voltage corresponding to logic 1 = 8 V R F = feedback resistance R 1 = input resistance for least significant input, 0.8 = 8 x R F R F = 0.1 R 1 R 1 R 1 = 10 x R F The following values satisfy this requirement: R F = 10 kω R 1 = 100 kω In order to get the correct weighting for the voltage gains for the four inputs, the input resistors are in the ratio R 1, R 1, R 1 and R The final circuit diagram is shown below: 197

29 The two resistors used in the second inverting amplifier can have any value, as long as both are of equal resistance (to give a voltage gain of -1) and have resistance greater than 1 kw (to reduce the size of the current flowing in them, and hence the power dissipated.) Exercise The diagram shows the circuit for a three-bit digital-to-analogue converter. In this system, logic 1 is 12 V and logic 0 is 0 V. (a) Calculate suitable values for the resistors so that the DAC has the following characteristics: Digital input Output voltage C B A V V R 1 = R 2 = R 3 = R 4 = 198

30 (b) The DAC is connected to a counter, as shown in the following diagram. Initially, the counter is reset. Then ten pulses are sent into the counter. Use the axes provided below to sketch the resulting output signal as this happens. 199

31 2. Design a DAC with the following characteristics: number of input bits = 3 output voltage range = 0 to 14 V logic 1 = 10 V and logic 0 = 0 V power supply = +15 V / 0 V / -15 V. The design must use the following components: two op-amps two 220 kω resistors one 20 kω resistor one 4 kω resistor two other resistors, for which you must calculate the resistance values. Circuit Diagram 200

32 Digitising audio signals Analogue recording media, such as magnetic tape, suffer from noise. Each time an analogue recording is copied, more noise is introduced and the quality is reduced - once noise is added to an analogue signal it cannot be removed. In digital recording, the analogue waveform is sampled at evenly-spaced time intervals. Each sample generates a binary number representing the amplitude of the analogue signal at that time. Digital recordings can be copied repeatedly without introducing additional noise. The quality of a digital recording depends on: Sample rate: must be more than twice the highest audio frequency present to allow the original signal to be reconstructed from the samples, (Nyquist sampling theory - considered in more detail in a later chapter.) Common sampling rates: 8 khz, (for speech signals) 44.1 khz (for music signals). Number of bits in a sample: as this increases: resolution increases quantization error decreases quality of the recording increases but amount of storage space increases. Audio CDs and most computer audio file formats use 16-bit. Other common bit formats are 12-bit, 24-bit and 32-bit. A block diagram of a typical digital record/playback system is shown below. Suppose that we want to record and playback audio signals up to a frequency of f khz. Recording: The cut off frequency of the low pass filter on the input (recording) section must also be f khz to remove any frequencies present that are too high for the chosen sampling rate. The sampling clock frequency must be greater than 2f khz. At each rising edge of the sampling clock pulse the sampling gate outputs the amplitude of the analogue signal at that instant. The ADC then converts each sample of the analogue information into an n-bit word. 201

33 The following image illustrates some signals obtained during this sampling process: Example: An analogue signal is sampled at a frequency of 30kHz and converted into a 14-bit binary code. The signal is recorded for 10 minutes. (a) Calculate the bit-rate and the file size of the digitized signal. Bit-rate = Sampling frequency x no. of bits per sample = 30 x 14 = 420 kbits (420 kbps) second File size = bit-rate x sampling time (in seconds) = 420 x 600 = kbits (252 Mbit) (b) The reference voltage of the ADC is 3 V. Determine the resolution of the ADC and the average quantization error. Resolution = i/p voltage range 2 n 14 3 = = 244 mv 2 Average quantization error = resolution 244 mv = = 122 mv 2 2 This example shows that quantization error is very small for large values of n. 202

34 Digital signal processor: This controls the format in which the audio files are stored. They can be stored uncompressed as.wav files or compressed, to save space, as.mp3 or.wma files for example. In addition, digital processing effects can be applied such as reverberation or pitch change. Playback: To playback the audio signal, the digital signal processor outputs each n-bit word, (fourteen bits in the example above), in parallel to the DAC. The DAC output is the staircase waveform described earlier. This is passed through the reconstruction filter, another low-pass filter, which smooths out the quantization steps as shown below. The sample rate has been increased to show the effect more clearly. Exercise For each of the following signals, determine the minimum sampling frequency required: Signal Frequency range Sampling frequency A 200 Hz to 4 khz B 100 Hz to 5 khz C 80 Hz to 6.8 khz D 120 Hz to 8.2 khz 203

35 2. A digital recording system uses a 24-bit encoding system. The maximum voltage range is 5 V. Calculate: (a) the resolution of the digitised signal (b) the average quantization error. 3. A recording studio creates a 74 minute compact disc using a 16-bit format and a sampling frequency of 44.1 khz. Calculate how much data is stored on the disc. 204