5.16 Digital signal processing problems  (Page 3/6)

Discrete-time filtering

We can find the input-output relation for a discrete-time filter much more easily than for analogfilters. The key idea is that a sequence can be written as a weighted linear combination of unit samples.

1. Show that $x(n)=\sum_i x(i)\delta (n-i)$ where $\delta (n)$ is the unit-sample. $$\delta (n)=\begin{cases}1 & \text{if $n=0$}\\ 0 & \text{otherwise}\end{cases}$$
2. If $h(n)$ denotes the unit-sample response —the output of a discrete-time linear, shift-invariant filter to a unit-sampleinput—find an expression for the output.
3. In particular, assume our filter is FIR, with the unit-sample response having duration $q+1$ . If the input has duration $N$ , what is the duration of the filter's output to thissignal?
4. Let the filter be a boxcar averager: $h(n)=\left(\frac{1}{q+1}\right)()$ for $n=\{0, \dots , q\}$ and zero otherwise. Let the input be a pulse of unit height and duration $N$ . Find the filter's output when $N=\frac{q+1}{2}$ , $q$ an odd integer.

A digital filter

A digital filter has the depicted unit-sample reponse.

1. What is the difference equation that defines this filter's input-output relationship?
2. What is this filter's transfer function?
3. What is the filter's output when the input is $\sin \left(\frac{\pi n}{4}\right)$ ?

A special discrete-time filter

Consider a FIR filter governed by the difference equation $$y(n)=\frac{1}{3}x(n+2)+\frac{2}{3}x(n+1)+x(n)+\frac{2}{3}x(n-1)+\frac{1}{3}x(n-2)$$

1. Find this filter's unit-sample response.
2. Find this filter's transfer function. Characterize this transfer function( i.e. , what classic filter category does it fall into).
3. Suppose we take a sequence and stretch it out by a factor of three. $$x(n)=\begin{cases}s(\frac{n}{3}) & \text{if $\forall m, m=\{\dots , -1, 0, 1, \dots \}\colon n=3m$}\\ 0 & \text{otherwise}\end{cases}$$ Sketch the sequence $x(n)$ for some example $s(n)$ . What is the filter's output to this input? In particular, what is the output at theindices where the input $x(n)$ is intentionally zero? Now how would you characterize this system?

Simulating the real world

Much of physics is governed by differntial equations, and wewant to use signal processing methods to simulate physical problems. The idea is to replace the derivative with adiscrete-time approximation and solve the resulting differential equation. For example, suppose we have thedifferential equation $$\frac{d y(t)}{d t}+ay(t)=x(t)$$ and we approximate the derivative by $$(t, , \frac{d y(t)}{d t})\approx \frac{y(nT)-y((n-1)T)}{T}$$ where $T$ essentially amounts to a sampling interval.

1. What is the difference equation that must be solved to approximate the differential equation?
2. When $x(t)=u(t)$ , the unit step, what will be the simulated output?
3. Assuming $x(t)$ is a sinusoid, how should the sampling interval $T$ be chosen so that the approximation works well?

Derivatives

The derivative of a sequence makes little sense, but still, we can approximate it. The digital filter described by the difference equation $$y(n)=x(n)-x(n-1)$$ resembles the derivative formula. We want to explore how well it works.

1. What is this filter's transfer function?
2. What is the filter's output to the depicted triangle input ?

   

3. Suppose the signal $x(n)$ is a sampled analog signal: $x(n)=x(n{T}_s)$ . Under what conditions will the filter act like a differentiator?In other words, when will $y(n)$ be proportional to $(t, , \frac{d x(t)}{d t})$ ?

The dft

Let's explore the DFT and its properties.

1. What is the length- $K$ DFT of length- $N$ boxcar sequence, where $N< K$ ?
2. Consider the special case where $K=4$ . Find the inverse DFT of the product of the DFTs of two length-3 boxcars.
3. If we could use DFTs to perform linear filtering, it should be true that the product of the input'sDFT and the unit-sample response's DFT equals the output's DFT. So that you can use what youjust calculated, let the input be a boxcar signal and the unit-sample response also be a boxcar. Theresult of part (b) would then be the filter's output if we could implement the filter with length-4 DFTs. Does the actualoutput of the boxcar-filter equal the result found in the previous part ?
4. What would you need to change so that the product of the DFTs of the input and unit-sampleresponse in this case equaled the DFT of the filtered output?