A stable LTI system is completely characterized in terms of its frequency response, H(jw). If X(jw) is the Continuous Ti

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A Stable Lti System Is Completely Characterized In Terms Of Its Frequency Response H Jw If X Jw Is The Continuous Ti 1 (108.41 KiB) Viewed 62 times

A stable LTI system is completely characterized in terms of its frequency response, H(jw). If X(jw) is the Continuous Time Frequency Transform (CTFT) of the system input, then, the output of system in the frequency domain is given by Y(jw) H(jw) X(jw). An LTI system can be defined by a differential equation, which represents the link between the input and output of the system. Let x(t) and y(t) be the input and output of the LTI system, respectively. In general, the differential equation used for representation of the such system is given by m {₁ d'xt) = 5 bd "xit) qu dtk dtm K:O m=0 The frequency response of the transfer function of the system is expressed by „M M-I by (jw) + bm (jw) + - +b, (jw) + bo Huw)= a (jw)+₁(jw) ~ +...+₁ (jus+a , where a, 0 sks N, and by, 0≤ b, sM, are real numbers. To determine the frequency response (including its magnitude and phase), "freqs* command of Matlab can be used. Consider an LTI system, which is defined as follows: dy(t). 3 2+ y(t) = -2 x(t) dt In Matlab, Its frequency response (for 0.01 sws 10) is determined by the following commands: a=(-2); b=[31]; w=-0.01:0.1:10; H-freqs(b,a,w); H_mag=abs(H); H_phase-angle(H)*180/pl; Note that H_mag and H_phase are magnitude and phase of the transfer function (H), respectively. To plot the phase and magnitude of the transfer function, the following commands can be applied subplot(211) loglog(w,H_mag); grid subplot(212) semilogx(w,H_phase) grid Task 1 1.1 Determine the frequency response of the LTI system defined by the below equation dy+ay(t)-ax(t), where a 4. de 1.2 Determine the impulse response of the system for the time domain Osts 10. For this purpose, you may apply "impulse" function of Matlab. 1.3 The abovementioned system can be expressed as a low-pass filter. To verify the quality of the proposed system, its performance in reduction of high frequency components is compared with the Butterworth filter, which is a well-known filter for continuous and discrete domains. To define the lowpass Butterworth filter for the continuous signal, the following commands are applied [ba]=butter(N,wc,'s'); Where "b" and "a" represents coefficients of the nominator and dominators, respectively. N is the order of the filter and "we" is the cut-off frequency of the filter (in angular frequency or rad/sec). Noe that if 's' is not used, the abovementioned Butterworth filter is defined for discrete signals. In the above definition, the transfer function of the filter is determined by T-tf(b,a); For the second-order lowpass Butterworth filter with wc-4, determine and plot its frequency response for 0.1 sw $10. Compare its results with the results obtained from the system defined in Section 1.1. What filter has a better performance in removal of high frequency components? Why? To answer this, provide graphs of results in one figure (for magnitude and phase, each separately). 1.4 Determine impulse responses of the Butterworth and first order filters defined in Sections 1.1 and 1,3. Plot their results and compare them with each other.. Task 2 In some practical filtering, it is required to have a non-negative output values from any non-negative input. This means for the output ylt), we have y(t)>0, -osts too, when x(t) 20 for all-costs +00. Based on the given impulse responses, what system satisfies this condition? Drive an example, whose output system can be negative, when the input is non-negative. Confirm your result by using "impulse" function of Matlab. Task 3 The input of an LTI system is given by x(t)=cos(7t)-2sin(2t) + cos(15t). It is required to extract "sin(2t)" component of the above signal to form the output of the system. This means, other components should be attenuated (or filtered). Design a minimum order and stable Butterworth filter satisfying the mentioned condition. Draw the input and output signals of system in the time and frequency domains. Draw the magnitude and phase of the designed filter. Hint: To calculate and plot Fourier transform of a signal (based on its definition), you may use "trapz" command of Matlab, which numerically integrates a function. Explore concept of this function by yourself and apply it in your program, if necessary.