Scaled unit impulse/step sequences

[freezer]

Homework Statement

[itex]b_{k} = {4, 1, 1, 4}[/itex]

x[n] = 2u[n]

Write your answer using scaled unit impulse sequences and scaled unit step sequences. Write explicitly.

Homework Equations

The Attempt at a Solution

4114
2222222...
------------
8228
 8228
  8228
   8228
    8228
      ...
------------
8, 10, 12, 20, 20, 20, ...

[itex]x[n] = 8\delta[n] + 10\delta[n-1] + 12\delta[n-2] + \sum^{\infty}_{k=3} 20 \delta[n-k][/itex][itex]
\delta[n] = \left\{\begin{matrix}
0 & n<0\\
8 & n = 0\\
10 & n = 1 \\
12 & n = 2\\
20 & n \geq 3
\end{matrix}\right.
[/itex]

Is this how you would answer this question?

 

[rude man]

You haven't stated the problem fully. What is the difference equation, exactly?


In any case, u[n] is always 1.

 

[freezer]

The general difference equation for a causal FIR is:

[itex]y[n] = \sum^{\infty}_{k=0} b_{k} x[n-k][/itex]

and then

[itex]
\delta[n] = \left\{\begin{matrix}
0 & n<0\\
8 & n = 0\\
10 & n = 1 \\
12 & n = 2\\
20 & n \geq 3
\end{matrix}\right.
[/itex]

 

[rude man]

Your expression for x[n] is correct. But I don't quite understand your table, probably because I can't make out the column after the 1st equal sign.

But δ[n] = 1, n = 0
= 0, n > 0 always.

BTW your equation is for a non-recursive filter, which is not necessarily an FIR filter.
FIR filters can also be recursive, and IIR filters can be non-recursive. However, saying FIR → non-recursive and IIR → recursive is almost universal, if misleading.

 

[rezeew]

I would respond by saying that the given sequence b_{k} can be expressed as a scaled unit impulse sequence and scaled unit step sequence. The scaled unit impulse sequence is represented by x[n] = 8\delta[n] + 10\delta[n-1] + 12\delta[n-2] + \sum^{\infty}_{k=3} 20 \delta[n-k], where \delta[n] represents the unit impulse function. This means that the sequence has a value of 8 at n=0, 10 at n=1, 12 at n=2, and 20 for all n greater than or equal to 3. The scaled unit step sequence is represented by x[n] = 2u[n], where u[n] represents the unit step function. This means that the sequence has a value of 2 for all n greater than or equal to 0. By combining these two representations, we can see that the given sequence b_{k} = {4, 1, 1, 4} can be expressed as 8\delta[n] + 10\delta[n-1] + 12\delta[n-2] + 20u[n-3]. This provides a more explicit representation of the sequence and allows for easier analysis and manipulation.