# Which one is a better impluse response by comparing two given filter length?

I was asked which impulse response $$h_1(t)$$ or $$h_2(t)$$ has a better decoding performance for two filter lengths: $$l_1 = 50$$ and $$l_2 = 500$$.

$$h_1(n) = \cfrac{1}{\sqrt{l_1}}\cos{(0.2\pi n f/f_s)}$$ $$h_2(n) = \cfrac{1}{\sqrt{l_2}}\cos{(0.2\pi n f/f_s)}$$ with $$0\leq n and $$f \in$$ [697 Hz , 1477 Hz]

The signal that I'm trying to decode is a discrete-time signal: $$x(t) = \cos{(2 \pi f_1 t)} + \cos{(2 \pi f_2 t)}$$ with sample length of 0.2s, sampling frequency $$f_s = 4000$$ Hz, $$f_1$$ and $$f_2 \in$$ [697 Hz , 1477 Hz] (you may recall that is the telephone number signal)

Q1) I would like to know if the sampling frequency of $$x(t)$$ is the same as the $$f_s$$ in $$h_1(n)$$ and $$h_2(n)$$.

Q2) I would like to know which of $$l_1$$ or $$l_2$$ values is better for encoding.

• Hi sss. I've edited your answer for formatting reasons, but you need to also edit it to reflect the claims you make. Specifically: - your impulse responses have no dependency on $t$ even though you write them $h(t)$ (they have a dependency on $n$ in your equations). - your $x(t)$ as written is a continuous-time signal, NOT discrete (where is $f_s$?). Keep in mind continuous signals are usually written as being dependent on time $t$, and discrete time signals are usually written as being dependent on sample number $n$. Once that's done I'm sure someone will be glad to answer!
– Jdip
Sep 13, 2022 at 6:31
• Thanks for pointing it out. h(t) should be h(n) instead.
– sss
Sep 13, 2022 at 6:59