Consider the following problems related to properties of filters.

(a) Filters that operate under real-time conditions need to be causal, i.e., they can only process present and past inputs. When no real-time processing is needed the filter can be non-causal.

i. Consider the case of averaging an input signal x[n] under real-time conditions. Suppose you are given two different filters

which of these would you use and why?

ii. If you are given a tape with the data which of the two filters would you use? Why? Would you use either? Explain.

(b) A significant difference between IIR and FIR discrete-time systems is stability. Consider an IIR filter with the difference equation y₁[n] = x[n] ˆ’ 0.5y₁[n ˆ’ 1] where x[n] is the input and y₁[n] is the output. Then consider an FIR filter

y₂[n] = x[n] + 0.5x[n ˆ’ 1] + 3x[n ˆ’ 2] + x[n ˆ’ 5]

where x[n] is the input and y₂[n] is the output.

i. Since to check the stability of these filters we need their impulse responses, find the impulse responses h₁[n] corresponding to the IIR filter by recursion, and h₂[n] corresponding to the FIR filter.

ii. Use the impulse responses h₁[n] and h₂[n] to check the stability of the IIR filter and of the FIR filter, respectively.

iii. Since the impulse response of an FIR filter has a finite number of non-zero terms, would it be correct to say that FIR filters are always stable? Explain.

Transcribed Image Text:

N-1 N-1 x[n – k] Уx[п — k], (В) (A) y[n] = k=0 2 (B) y[n] = k=-N+1