Question

The signal corresponding to the sinusoid x(t) = 10sin(2π * 5000t - π/2) is sampled at the frequency Fe = 44100 samples/second, after having been previously filtered by an anti-aliasing analog filter whose positive phase shift at 5000 Hz is 1 radian. The digitized signal obtained is x[n]. We then filter x[n] with an FIR filter whose difference equation is y[n] = x[n] - 0.9x[n-1] + 1.2x[n-3]. Give the equation of y[n] obtained at the output of this filter, in steady state. You will first need to determine the gain and phase of the filter at the frequency of interest.

Answer 1

The equation of y[n] obtained at the output of the FIR filter in steady state is y[n] = gain * x[n] * cos(phase). The gain and phase can be calculated using the difference equation and the given sinusoid. The gain is positive and there is a phase shift which can be determined as the angle of the complex result.

Step-by-step explanation:

To determine the equation of y[n] at the output of the FIR filter, we first need to find the gain and phase of the filter at the frequency of interest. The frequency of interest is 5000 Hz. Given the difference equation y[n] = x[n] - 0.9x[n-1] + 1.2x[n-3], we can substitute x[n] with the given sinusoid x(t) = 10sin(2π * 5000t - π/2).

We can calculate the gain of the filter by substituting the values of x[n] and x[n-1] into the difference equation. The gain is positive, so there is no inversion of the signal. To calculate the phase shift, we substitute the values of x[n] and x[n-1] into the difference equation. The phase shift is given by the angle of the complex result of this calculation.

In steady state, the equation of y[n] obtained at the output of this filter is y[n] = gain * x[n] * cos(phase), where the gain is the calculated value and the phase is the calculated phase shift.