 achieve than Obviously, the closer the spacing of the discrete levels, the smaller conversion from analogue to digital. There are again many tech- will be the quantisation distortion. However, the smaller the spac- niques which may be used for this conversion, each possessing its ing, the greater is the number of levels. This increases the number own blend of merits and demerits. of digits in the code words representing the levels (e.g. doubling the The simplest form of DAC is shown in Figure 19.18. It consists of number of levels increases the length of a binary code word by one an array of resistors connected via an analogue switch either to a bit). An increase in the number of digits representing every sample reference voltage, if the corresponding bit in the digital word is increases the bandwidth required to transmit them. 'one', or to ground otherwise. The resistors are all connected to the Consider a quantiser where the ith interval is centred around Q virtual earth input of a current summing amplifier, and their values and possesses width Any input signal X in the range given by are chosen such that the currents flowing through the resistors when the corresponding bits are 'one' are binary weighted. Equation 19.44, will be represented by the quantised amplitude O (19.44) The squared quantisation error will clearly be (x-Q)2. us also assume that the probability density of the input signal is p(x). Therefore, to obtain the mean square error over the ith range, the squared error value must be averaged over all possible values of the input signal for that quantisation interval, as in Equation 19.45. Analogue (19.45) output If the signal varies widely in comparison to the size of a single quantisation interval, it can be assumed that p(x) is constant across that quantisation interval, having value Therefore Equation 19.46 is obtained. Figure 19.18 Simple digital to analogue converter (19.46) The output voltage of the summing amplifier is proportional to the sum of the input currents, and hence to the numerica