5.16 Digital signal processing problems  (Page 4/6)

Dsp tricks

Sammy is faced with computing lots of discrete Fourier transforms. He will, of course, use the FFT algorithm, but he is behind schedule and needsto get his results as quickly as possible. He gets the idea of computing two transforms at one time by computing the transform of $s(n)=s_1(n)+i{s}_2(n)$ , where $s_1(n)$ and $s_2(n)$ are two real-valued signals of which he needs to compute the spectra. The issue is whether he can retrievethe individual DFTs from the result or not.

1. What will be the DFT $S(k)$ of this complex-valued signal in terms of $S_1(k)$ and $S_2(k)$ , the DFTs of the original signals?
2. Sammy's friend, an Aggie who knows some signal processing, says that retrieving the wanted DFTs is easy:“Just find the real and imaginary parts of $S(k)$ .” Show that this approach is too simplistic.
3. While his friend's idea is not correct, it does give him an idea. What approach will work? Hint : Use the symmetry properties of the DFT.
4. How does the number of computations change with this approach? Will Sammy's idea ultimately lead to afaster computation of the required DFTs?

Discrete cosine transform (dct)

The discrete cosine transform of a length- $N$ sequence is defined to be $$S_c(k)=\sum_{n=0}^{N-1} s(n)\cos \left(\frac{2\pi nk}{2N}\right)$$ Note that the number of frequency terms is $2N-1$ : $k=\{0, \dots , 2N-1\}$ .

1. Find the inverse DCT.
2. Does a Parseval's Theorem hold for the DCT?
3. You choose to transmit information about the signal $s(n)$ according to the DCT coefficients. You could only send one, which one would you send?

A digital filter

A digital filter is described by the following difference equation: $$y(n)=ay(n-1)+ax(n)-x(n-1)\text{,}a=\frac{1}{\sqrt{2}}$$

1. What is this filter's unit sample response?
2. What is this filter's transfer function?
3. What is this filter's output when the input is $\sin \left(\frac{\pi n}{4}\right)$ ?

Another digital filter

A digital filter is determined by the following difference equation. $$y(n)=y(n-1)+x(n)-x(n-4)$$

1. Find this filter's unit sample response.
2. What is the filter's transfer function? How would you characterize this filter (lowpass, highpass, special purpose, ...)?
3. Find the filter's output when the input is the sinusoid $\sin \left(\frac{\pi n}{2}\right)$ .
4. In another case, the input sequence is zero for $n< 0$ , then becomes nonzero. Sammy measures the output to be $y(n)=\delta (n)+\delta (n-1)$ . Can his measurement be correct?In other words, is there an input that can yield this output? If so, find the input $x(n)$ that gives rise to this output. If not, why not?

Yet another digital filter

A filter has an input-output relationship given by the difference equation $$y(n)=\frac{1}{4}x(n)+\frac{1}{2}x(n-1)+\frac{1}{4}x(n-2)$$ .

1. What is the filter's transfer function? How would you characterize it?
2. What is the filter's output when the input equals $\cos \left(\frac{\pi n}{2}\right)$ ?
3. What is the filter's output when the input is the depicted discrete-timesquare wave ( [link] )?

A digital filter in the frequency domain

We have a filter with the transfer function $$H(e^{i\times 2\pi f})=e^{-(i\times 2\pi f)}\cos (2\pi f)$$ operating on the input signal $x(n)=\delta (n)-\delta (n-2)$ that yields the output $y(n)$ .

1. What is the filter's unit-sample response?
2. What is the discrete-Fourier transform of the output?
3. What is the time-domain expression for the output?

Digital filters

A discrete-time system is governed by the difference equation $$y(n)=y(n-1)+\frac{x(n)+x(n-1)}{2}$$

1. Find the transfer function for this system.
2. What is this system's output when the input is $\sin \left(\frac{\pi n}{2}\right)$ ?
3. If the output is observed to be $y(n)=\delta (n)+\delta (n-1)$ , then what is the input?