Frequency Response
Consider numerical differentiation seen in Assignment #1 and used in an article appearing in The American Journal of Cardiology, 1991; 67: 965-975.
“Each sampled signal was differentiated according to the formula
y[n] = .5{x[n + 1] − x[n − 1]} + (1/8){x[n + 2] − x[n − 2]}
This differentiation has the convenient characteristic of shifting the frequency content of the signal upward in a precise mathematical way.”Using linear systems concepts, you can quantify the “precise mathematical way.”
(a) Find the the frequency response, H(e^jω).
(b) Is this discrete time system a good approximation for differentiating a continuous-time signal?For some inputs? For all inputs? For no inputs? Explain your claim.
Hint: In the frequency domain, consider both the magnitude and phase response; recall that d dtx(t) has Fourier transform jωX(jω). Let sampling period be T seconds.
(c) How would you make this system causal? How, precisely, would your suggested change affect the frequency response? Frequency Response
Consider numerical differentiation seen in Assignment #1 and used in an article appearing in The American Journal of Cardiology, 1991; 67: 965-975.
"Each sampled signal was differentiated according to the formula
y[n] = .5{x[n + 1] - x[n - 1]} + (1/8){x[n + 2] - x[n - 2]}
This differentiation has the convenient characteristic of shifting the frequency content of the signal upward in a precise mathematical way."
Using linear systems concepts, you can quantify the "precise mathematical way." (a) Find the the frequency response, H(e^jω).
(b) Is this discrete time system a good approximation for differentiating a continuous-time signal?
For some inputs? For all inputs? For no inputs? Explain your claim.
Hint: In the frequency domain, consider both the magnitude and phase response; recall that d dtx(t) has Fourier transform jωX(jω). Let sampling period be T seconds.
(c) How would you make this system causal? How, precisely, would your suggested change affect the frequency response?