LIR Filter for Single Sinusoid Generation - Essay Example

Lab report INTRODUCTION The main aim of the experiment is to aids in comprehending design and execution of the IIR filters using MATLAB and the C6713 system. Difference equations is important because of the existence of the IIR filters that possess an extended impulse response {h(n)} and the use of convolution for filtering the input signal is not recommended. Therefore, the difference equations are used to perform this operation. Also, the pole-zero plots are significant in the analysis of a signal with regards to the position of the poles by indicating whether the signal is stable or unstable PROCEDURE Experiment 1: IIR filter for single sinusoid generation The exercise involved the use of a sinusoidal signal generated in MATLAB for (preliminary Q.5) generated for this experiment using the C6713 system board with f (frequency) = 500 Hz, Fs (sampling frequency) = 8 kHz and A (amplitude) = 1V. IIR filter was applied to this signal with different kind of equations. A construction of a pole-zero plot was applied with the aim of understanding the nature and functions of the signals. Experiment 2: Implementing a Butterworth IIR lowpass filter on C7613 For this experiment, a fourth order Butterworth IIR filter was to be designed on MATLAB and C6713 using the following difference equation: The 3-dB cut-off frequency of the filter 1.5 kHz and the sampling frequency was 8 kHz. To design this 4-th order low-pass filter on MATLAB, the butter function was used: [b,a]= butter(4,wn,low); where wn is the cut-off frequency divided by half the sampling frequency =(  RESULTS Figure 3: Signal generation on C7613 system Figure 4: Poles-Zeroes Diagram for IIR filter Figure 5: Magnitude function |H (w)| (linear and dB plot / Note: Linear plot (top) & dB plot (bottom) Table 1: filter cofficients Filter coefficients [b, a] Table1: Filter coefficients [b, a] b= 0.037973 0.151892 0.227838 0.151892 0.037973 a= 1.000000 -0.978369 0.790086 -0.241882 0.037734 Figure 6: Pole-zero plot for Butterworth filter Figure 6: Pole-zero plots for Butterworth filter Figure 7: Phase response of < H (w) For the implementation of the filter on the C6713, the magnitude function was swept at different frequencies to see the effect on the Vout (output voltage). Figure 8 and 9 show the effect of sweeping frequency for one at 1000 Hz and the other at 2000 Hz. Figure 8: Frequency at 1000 Hz Figure 9: Frequency at 2000 Hz Table 2: Voltage Amplitude for Input/output Table2: The gain for different frequency Frequency (Hz) Vpp (IN) (V) Vpp (OUT) (V) Gain 250 1 1.04 1.04 500 1 1.04 1.04 1000 1 1.04 1.04 1100 1 1.02 1.02 1200 1 1 1 1300 1 0.92 0.92 1400 1 0.86 0.86 1500 1 0.7 0.7 1600 1 0.6 0.6 1700 1 0.52 0.52 1800 1 0.4 0.4 1900 1 0.28 0.28 2000 1 0.24 0.24 2100 1 0.2 0.2 2200 1 0.14 0.14 2300 1 0.12 0.12 2400 1 0.08 0.08 2500 1 0.08 0.08 The gain was determined through using the equation: Vout=output voltage (V), Vin=input voltage (V) and A=amplitude. The magnitude function of the filter created in MATLAB and C6713 system were then plotted on MATLAB for comparison, as shown in Figure 10. Figure 10: Overlaid MATLAB plot of magnitude function(s) Note: red line (designed Butterworth filter MATLAB) and blue line (filter implemented on C6713 system). DISCUSSION Experiment 1 majored on the application of an IIR filter to generate a single sinusoid on C6713 system. As illustrated in the above experiment, the different equations were used in the construction of the sinusoidal signal and a pole-zero plots was used in making observations. The pole-zero plot in figure 4 shows two poles located on the unit circle; this implies that the signal is ‘marginally stable’ meaning that it maintains its stability throughout the experiment (Kuo, Lee & Tian, 2013). This is the trend for the waveform evidenced in Figure 3, where the signal remains constant as it oscillates. This is contrary if the poles were located within the unit circle, the signal would change by beginning to decay as it is considered stable. If the poles were located outside circle, the signal would start increasing, which implies that the signal is unstable. For the case in experiment 2, a 4th order Butterworth filter designed both in MATLAB and on the C6713 system. For the Butterwort filter in MATLAB portion, the magnitude response (linear and dB) was constructed while conditioned on the C6713 system, the voltage inputs and outputs were measured at different frequencies. A construction of a pole-zero plots was made as a feedback to the magnitude response. In doing so, Figure 6 shows the poles are located inside the unit circle which elaborates that the signal is decaying and is evidently illustrated with the magnitude response (linear) of the filter (Kuo, Lee & Tian, 2013). Figure 10, consisting of both plots, show that the gain decreases as begins to decrease while sweeping at higher frequencies. In Table 1, the output voltage is 1.04 V for 250 Hz, but the onset of the decrease begins when the frequency increases hence the amplitude of waveform becomes smaller. The group delay indicates whether there will be phase distortions depending on the phase response of the signal. In this case, the phase response as illustrated in Figure 7 displays a small linear trend though, since it is not exactly linear, there will be some phase distortions in the signal. A linear phase response means no phase distortions are present while a non-linear phase response will cause the signal to have phase distortions. Conclusion The experiments use different equations in the construction of the sinusoidal signal and a pole-zero plots was used in making observations. The pole-zero shows two poles located on the unit circle depicting es that the signal is ‘marginally stable’ meaning that it maintains its stability throughout the experiment. Contrary, if the poles were located within the unit circle, the signal would change by beginning to decay as it is considered stable. If the poles were located outside circle, the signal would start increasing, which implies that the signal is unstable. The application of difference equations is significant due to the fact that IIR filters have an extended impulse response {h(n)} and the use of convolution for filtering the input signal is not recommended. Therefore, the difference equations are used to perform this operation. Also, the pole-zero plots are significant in the analysis of a signal with regards to the position of the poles by indicating whether the signal is stable or unstable Reference Kuo, S. M., Lee, B. H., & Tian, W. (2013). Real-time digital signal processing: Fundamentals, implementations and applications. Read More