4.10 Frequency domain problems  (Page 2/6)

In this problem, we want to understand the temperature component of our environment using Fourier series andlinear system theory. The file temperature.mat contains these data (daylight hours in the first row, corresponding averagedaily highs in the second) for Houston, Texas.

1. Let the length of day serve as the sole input to asystem having an output equal to the average daily temperature. Examining the plots of input andoutput, would you say that the system is linear or not? How did you reach you conclusion?
2. Find the first five terms ( $c_0()$ , ... , $c_4()$ ) of the complex Fourier series for each signal.Use the following formula that approximates the integral required to find the Fourier coefficients. $$c_k=\frac{1}{366}\sum_{n=0}^{366} s(n)e^{-(i\frac{2\pi nk}{366})}$$
3. What is the harmonic distortion in the two signals? Exclude $c_0()$ from this calculation.
4. Because the harmonic distortion is small, let's concentrate only on the first harmonic. What is thephase shift between input and output signals?
5. Find the transfer function of the simplest possible linear model that would describe the data.Characterize and interpret the structure of this model. In particular, give a physical explanationfor the phase shift.
6. Predict what the output would be if the model had no phase shift. Would days be hotter? If so, by howmuch?

Fourier transform pairs

Find the Fourier or inverse Fourier transform of the following.

1. $\forall t\colon x(t)=e^{-(a\left|t\right|)}$
2. $x(t)=te^{-(at)}u(t)$
3. $X(f)=\begin{cases}1 & \text{if $\left|f\right|< W$}\\ 0 & \text{if $\left|f\right|> W$}\end{cases}$
4. $x(t)=e^{-(at)}\cos (2\pi f_{0}t)u(t)$

Duality in fourier transforms

"Duality" means that the Fourier transform and the inverse Fourier transform are very similar. Consequently, the waveform $s(t)$ in the time domain and the spectrum $s(f)$ have a Fourier transform and an inverse Fourier transform, respectively, that are very similar.

1. Calculate the Fourier transform of the signal shown below .
2. Calculate the inverse Fourier transform of the spectrum shown below .
3. How are these answers related? What is the general relationship between the Fourier transform of $s(t)$ and the inverse transform of $s(f)$ ?

Spectra of pulse sequences

Pulse sequences occur often in digital communication and in other fields as well. What are their spectral properties?

1. Calculate the Fourier transform of the single pulse shown below .
2. Calculate the Fourier transform of the two-pulse sequence shown below .
3. Calculate the Fourier transform for the ten -pulse sequence shown in below . You should look for a general expression that holds for sequences of any length.
4. Using Matlab, plot the magnitudes of the three spectra. Describe how the spectra change as the number of repeated pulses increases.

Spectra of digital communication signals

One way to represent bits with signals is shown in [link] . If the value of a bit is a “1”, it is represented by a positive pulse of duration $T$ . If it is a “0”, it is represented by a negative pulse of the same duration.To represent a sequence of bits, the appropriately chosen pulses are placed one after the other.

1. What is the spectrum of the waveform that represents the alternating bit sequence “...01010101...”?
2. This signal's bandwidth is defined to be the frequency range over which 90% of the power is contained. What is this signal's bandwidth?
3. Suppose the bit sequence becomes “...00110011...”. Now what is the bandwidth?