Determine the requirements of the digital fiters needed and

Question

(1) Assume that we are given the continuous-time signal

f₁ = 5 Hz, f₂ = 12 Hz, f₃ = 5 Hz, and f₄ = 35 Hz, f₅ = 60 Hz, and f₆ = 100 Hz. It is required to design a digital signal processing-based system to separated the signals xa₁(t) and xa₂(t) from the signal xa(t). Assuming a sampling frequency of 200 samples/second, the order of the fiters to be n = 6, and an attenuation not more that 1 dB in the passband for the used digital fiters.

(a) Determine the requirements of the digital fiters needed.

(b) Design and write the dierence equations for the digital fiters.

(c) Plot the attenuation (in dB) response of each fiter to verify your design.

(d) Draw the complete block diagram of the system giving the requirements of the other components of the system needed in conjunction with the digital fiters.

(2) Consider the continuous-time systems described by the following transfer functions:

(a) Assuming that the above 2 systems are connected in series, nd the composite transfer function Ha(s). Hence, nd the poles and zeros. Is the fiter stable?

(b) Plot the magnitude response in versus the frequency in Hz. Hence, nd the frequency at which the magnitude is zero. Call this frequency fz Hz. Hint: consider the frequency range 0 ≤ f ≤ 20 Hz.

(c) It is required to replace the above fiter by a digital fiter counterpart such that the magnitude response is zero at the same frequency fz Hz, assuming that the sampling rate is 100 samples/second. Using the bilinear transformation, determine the digital fiter transfer function H(z).

(d) Plot the magnitude response of the digital fiter versus f in Hz to verify your results.

(e) Draw the complete diagram of the system showing the requirements of the other needed components.

(3) It is required to design a second order digital lowpass-notch fiter with the following specications:

 A sampling frequency of 20 kHz and a dc frequency response of 22.5. Frequency response of 10 at a frequency of f = 10 kHz.

 A frequency response of zero at a frequency of f = 3 kHz.

The digitallowpass-notch fiter is obtained from a second order analog lowpass-notch fiter through the application of the bilinear transformation, where the analog fiter transfer function is given by:

(a) From the above mentioned facts and the relation between, the analog frequency, and its digital counterpart frequency !, determine the values of z, the parameter k and the bilinear parameter T:

(b) Determine the transfer function of the analog fiter. Hence, plot the magnitude response. Verify the analog lowpass-notch parameters.

(c) Design the corresponding digital lowpass-notch fiter from the analog using the bilinear transformation.

(d) Plot the magnitude of the digital (freq. in kHz) to verify your results.

(e) Write the corresponding dierence equation.

(4) It is required to design a second order digital allpass fiter with the following specications:

A constant magnitude response of 5.

A sampling frequency of 100 kHz

A phase response of 40o at a frequency of f = 12:9 kHz

A phase response of 100o at a frequency of f = 17:55 kHz.

The digital allpass fiter is obtained from a normalized second order analog allpass fiter through the application of the bilinear transformation, where the analog fiter transfer function is given by:

(a) Derived expressions for the magnitude and phase responses for the analog fiters in terms of k and Q.

(b) From the above mentioned facts and the relation between , the analog frequency, and its digital counterpart frequency !, design the digital allpass fiter. Hence, write the corresponding dierence equation.

(c) Plot the magnitude and phase responses (freq. in Hz) to verify your results.

(d) Plot the group delay.