Assignment -
Problem 1 - Given is the LCC difference equation that represents some LTI system:
y(n) - ¾y(n-1) - ¼y(n-2) = x(n) + x(n-1)
a) Find the impulse response of the system (solve the LCCDE).
b) Draw a block diagram of the system in Direct Form I.
c) Is the system stable? Justify!
Problem 2 - Given is the LCC difference equation that represents an LTI system:
y(n) - y(n-1) + ¼y(n-2) = 2x(n)
a) Find the impulse response of the system. Is the system stable?
b) Draw a block diagram of the system in Direct Form II.
c) Find the response of system to the input x(n) = 4(1/2)nu(n) and assume y(-1) = 2, y(-2) = 8.
Problem 3 - In many DSP applications, it is necessary to estimate (and subsequently remove) the DC bias from the input signal. Given is a C-function (not the best code example) and the Matlab equivalent to estimate the DC component of a signal. The dc_est is initialized and the function is called repeatedly for each new sample of the input signal. Each time it is called, the function receives the current value of the input signal (x_n) and returns the current estimate of the DC value of that signal. It takes some time before the dc_est converges to the true DC value. We can view the function as an LTI system with constant coefficients, with input x(n) and output y(n) (equal to the current value of dc_est).
/* C- FUNCTION */
#define ALPHA 0.001
double dc_est = 0; /* global dc-estimate initialized to 0 */
double dc_estimator(double x_n)
{
dc_est = dc_est + ALPHA *(x_n - dc_est);
return dc_est;
}
EQUIVALENT MATLAB FUNCTION
function dc_est = dc_estimator( x_n )
global dc_est % initialized to 0 prior to calling the function for the first time
ALPHA = 0.001;
dc_est = dc_est + ALPHA*(x_n - dc_est);
end
a) Write the difference equation that corresponds to the DC estimator system.
b) Assuming input signal x(n) = 5 u(n), solve the difference equation and provide the analytical expression for the output in terms of parameter ALPHA. (Note: the initial condition is given above).
c) Assuming x(n) = 5 u(n) and ALPHA = 0.001, how many samples n will it take for the output i.e. "dc_est" to be within 1% of the actual DC value? If the sampling rate is 48000 Hz, how much time (in milliseconds) will take the output to be within 1% of the DC value?
d) Repeat part c) for ALPHA=0.01.
e) Roughly sketch or plot the output of the dc_estimator system to the input in part c) for ALPHA=0.001 and for ALPHA=0.01 Draw a block diagram of the system in Direct II form.
f) Let x(n) = 10 sin(w0n)u(n), where w0 = π/2. What is the DC component of this signal (for n ≥ 0)? Assume we feed this signal into the dc_estimator system and let enough time pass so that the steady-state is reached i.e. the homogeneous part of the response of the system is no longer present. Thus, for this part, it is only necessary to find the particular solution of the equation.
Questions for part f):
Find the particular solution to the sinusoidal input.
How do we know that homogenous response will no longer be present for large enough n?
What is the maximum deviation of the dc_estimator response from the true DC value?
What happens to this maximum deviation when ALPHA=0.001 vs ALPHA=0.01.