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EL871 Laboratory 14/10/2018 EL871. Continuous Assessment Laboratory 1 Introduction This problem sheet is the first of 4 assessed laboratories for the EL871 Module. The previous examples sheet and associated lectures will have prepared you for the problems set. As this is assessed work the demonstrators are only there to help with any problems with the software (MATLAB) or equipment. They are not supposed to help you answer the questions. There are 3 questions answer any 2 questions. Work should be submitted as a report to the reception by 12:00 pm on Monday 21th October 2019 and MOODLE. Any MATLAB code used should also be included in the report for assessment. Solutions must be in a single file in word format. 1. An analog signal with a uniform power spectral density is bandlimited by a Butterworth filter with the following amplitude response:   1 2 4 1 1 c H f f f             Where fc = 4 kHz. The signal is digitised using a 14-bit ADC. Determine the minimum sampling frequency so that the maximum aliasing error is less than the quantisation error level in the passband. Confirm your solution by plotting the filter characteristics in MATLAB. Your solution should include your calculations (and any assumptions you have made) and the MATLAB plots. [10 marks] 2. The requirements for the analog input section of a real-time DSP system are: Frequency band of interest is 0 – 8 kHz, i.e. fc=8KHz Maximum permissible passband ripple 0.5 dB Stopband Attenuation 50 dB . Determine the minimum order of an anti-aliasing filter with Butterworth characteristics and a suitable sampling frequency to satisfy the requirements. Your solution should include a plot of the filter characteristic, all working and any EL871 Laboratory 14/10/2018 assumptions you have made in obtaining your solution. Your calculation should be verified using MATLAB. Hint: Previous examples have used the -3dB point as the end of the passband (when the signal magnitude is 0.7071 what it is at 0 Hz). This example expects the reference point to be at -0.5 dB. The equation for a Low-Pass Butterworth filter response is monotonically decreasing from 0 Hz. [10 marks] 3. An audio system handles signals with a baseband that extends from 0 to 20 kHz. It has been decided to use an oversampled 12-bit ADC to achieve 16-bit performance. Determine the minimum sampling frequency that would be required for a 12-bit converter and hence calculate the oversampling frequency required for the 12-bit converter to achieve 16-bit performance. [10 marks] You may assume the following: The intrinsic quantisation noise power, introduced by an ADC is given by  2 12 2 2 12 12 B q      (normalised) Where B is the ADC wordlength (Number of bits). For sufficiently large or random analog input signals, the energy of the quantisation noise is spread evenly over the available spectrum, i.e. from 0 to 2sF where sF is the sampling frequency. In this case the power spectral density of the quantisation noise  eP f is given by   2 e s P f F   The effective resolution of the ADC can be increased by sampling the input data at a high rate to “spread” the quantisation noise energy over a wider frequency band, thereby reducing noise levels in the band of interest as illustrated in Figure 1. This is what is meant by oversampling. At the Nyquist rate (i.e. max2sF f ) the normalised in-band quantisation noise for the 12-bit and 16-bit converters are, respectively:  12 1 2 1 2 12 B     where B1 = 12 EL871 Laboratory 14/10/2018  12 1 2 2 2 12 B     where B2 = 16 -Fs/2 Fs/20 Pe ( f ) 2 sF  -Fs/2 Fs/20 Pe ( f ) 2 sF  (a) (b) Figure 1. Quantisation noise power spectral density (a) Nyquist rate converter and (b) oversampled converter. The total noise power is the same for both converters, but for the oversampled converter the noise power is distributed over a much wider frequency range leading to a smaller in-band noise power level If the in-band quantisation noise power is reduced by the oversampling factor as described by the equation: 2 2max 1 1 2 s f F    You can then use this equation to determine the oversampling factor and therefore the oversampling frequency. Your solution should include all working and any assumptions you have made.


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