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Fourier Series Problems

Module by: Don Johnson

Summary: (Blank Abstract)

Problem 1:

Fourier Series

Find the Fourier series representation for the following periodic signals. For signal (c), find the complex Fourier series for the triangle wave without performing the usual Fourier integrals. Hint: How is this signal related to one for which you already have the series?

Figure 1
signal asignal bsignal c
Subfigure 1.1Subfigure 1.2Subfigure 1.3
signal a (sig6.png)signal b (sig7.png)signal c (sig8.png)
No answer provided.
Problem 2:

Phase Distortion

We can learn about phase distortion by returning to circuits and investigate the following circuit.

Figure 2
Figure 2 (circuit3.png)
  1. Find this filter's transfer function.
  2. Find the magnitude and phase of this transfer function. How would you characterize this circuit?
  3. Let v in t v in t be a square-wave of period TT. What is the Fourier series for the output voltage?
  4. Use Matlab to find the output's waveform for the cases T=0.01 T 0.01 and T=2 T 2 What value of TT delineates the two kinds of results you found? The software in fourier2.m might be useful.
  5. Instead of the depicted circuit, the square wave is passed through a system that delays its input, which applies a linear phase shift to the signal's spectrum. Let the delay τ be t4 t 4 . Use the transfer function of a delay to compute, using Matlab, the Fourier series of the output. Show that the square wave is indeed delayed.
No answer provided.
Problem 3:

Approximating Periodic Signals

Often, we want to approximate a reference signal by a somewhat simpler signal. To assess the quality of an approximation, the most frequently used error measure is the mean-squared error. For a periodic signal st s t , ε2=1T∫0Tst- s ˜ t2dt ε 2 1 T t 0 T s t s ˜ t 2 where st s t is the reference signal and s ˜ t s ˜ t its approximation. One convenient way of finding approximations for periodic signals is to truncate their Fourier series. s ˜ t=∑k=-KK c k ⅇⅈ2πkTt s ˜ t k -K K c k 2 k T t The point of this problem is to analyze whether this approach is the best (i.e., always minimizes the mean-squared error).

  1. Find a frequency-domain expression for the approximation error when we use the truncated Fourier series as the approximation.
  2. Instead of truncating the series, let's generalize the nature of the approximation to including any set of 2K+1 2 K 1 terms: We'll always include the c0 c0 and the negative indexed term corresponding to ck ck . What selection of terms minimizes the mean-squared error? Find an expression for the mean-squared error resulting from your choice.
  3. Find the Fourier series for the depicted signal. Use Matlab to find the truncated approximation and best approximation involving two terms. Plot the mean-squared error as a function of KK for both approximations.

Figure 3
Figure 3 (sig9.png)
No answer provided.
Problem 4:

Long, Hot Days

The daily temperature is a consequence of several effects, one of them being the sun's heating. If this were the dominant effect, then daily temperatures would be proportional to the number of daylight hours. The plot shows that the average daily high temperature does not behave that way.

Figure 4
Figure 4 (sig38.png)

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

  1. Let the length of day serve as the sole input to a system having an output equal to the average daily temperature. Examining the plots of input and output, 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 0 ,..., c 4 c 4 ) of the complex Fourier series for each signal.
  3. What is the harmonic distortion in the two signals? Exclude c 0 c 0 from this calculation.
  4. Because the harmonic distortion is small, let's concentrate only on the first harmonic. What is the phase 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 explanation for the phase shift.
  6. Predict what the output would be if the model had no phase shift. Would days be hotter? If so, by how much?

No answer provided.
Problem 5:

Fourier Transform Pairs

Find the Fourier or inverse Fourier transform of the following.

  1. xt=ⅇ-a|t| x t -a t for -∞<t<∞ t
  2. xt=tⅇ-atut x t t -a t u t
  3. Xf=1if|f|<W0if|f|>W X f 1 f W 0 f W
  4. xt=ⅇ-atcos2πf0tut x t a t 2 f0 t u t

No answer provided.
Problem 6:

Lowpass Filtering a Square Wave

Let a square wave (period TT) serve as the input to a first-order lowpass system constructed as a RCRC filter. We want to derive an expression for the time-domain response of the filter to this input.

  1. First, consider the response of the filter to a simple pulse, having unit amplitude and width T2 T 2 . Derive an expression for the filter's output to this pulse.
  2. Noting that the square wave is a superposition of a sequence of these pulses, what is the filter's response to the square wave?
  3. The nature of this response should change as the relation between the square wave's period and the filter's cutoff frequency change. How long must the period be so that the response does not achieve a relatively constant value between transitions in the square wave? What is the relation of the filter's cutoff frequency to the square wave's spectrum in this case?

No answer provided.
Problem 7:

Mathematics with Circuits

Simple circuits can implement simple mathematical operations, such as integration and differentiation. We want to develop an active circuit (it contains an op-amp) having an output that is proportional to the integral of its input. For example, you could use an integrator in a car to determine distance traveled from the speedometer.

  1. What is the transfer function of an integrator?
  2. Find an op-amp circuit so that its voltage output is proportional to the integral of its input for all signals.

No answer provided.
Problem 8:

Where is that sound coming from?

We determine where sound is coming from because we have two ears and a brain. Sound travels at a relatively slow speed and our brain uses the fact that sound will arrive at one ear before the other. Here, a sound coming from the right arrives at the left ear ττ seconds after it arrives at the right ear.

Figure 5
Figure 5 (sound.png)

Once the brain finds this propagation delay, it can determine the sound direction. In an attempt to model what the brain might do, RU signal processors want to design an optimal system that delays each ear's signal by some amount then adds them together. Δl Δl and Δr Δr are the delays applied to the left and right signals respectively. The idea is to determine the delay values according to some criterion that is based on what is measured by the two ears.

  1. What is the transfer function between the sound signal st s t and the processor output yt y t ?
  2. One way of determining the delay ττ is to choose Δl Δl and Δr Δr to maximize the power in yt y t . How are these maximum-power processing delays related to ττ?

No answer provided.
Problem 9:

Arrangements of Systems

Architecting a system of modular components means arranging them in various configurations to achieve some overall input-output relation. For each of the following, determine the overall transfer function between xt x t and yt y t .

Figure 6
system asystem bsystem c
Subfigure 6.1Subfigure 6.2Subfigure 6.3
system a (sys1.png)system b (sys2.png)system c (sys3.png)

The overall transfer function for part (a) is particularly interesting. What does it say about the effect of the ordering of linear, time-invariant systems in a cascade?

No answer provided.
Problem 10:

Reverberation

Reverberation corresponds to adding to a signal its delayed version.

  1. Assuming ττ represents the delay, what is the input-output relation for a reverberation system? Is the system linear and time-invariant? If so, find the transfer function; if not, what linearity or time-invariance criterion does reverberation violate.
  2. A music group known as the ROwls is having trouble selling its recordings. The record company's engineer gets the idea of applying different delay to the low and high frequencies and adding the result to create a new musical effect. Thus, the ROwls' audio would be separated into two parts (one less than the frequency f0 f0 , the other greater than f0 f0 , these would be delayed by τl τl and τh τh respectively, and the resulting signals added. Draw a block diagram for this new audio processing system, showing its various components.
  3. How does the magnitude of the system's transfer function depend on the two delays?

No answer provided.
Problem 11:

Echoes in Telephone Systems

A frequently encountered problem in telephones is echo. Here, because of acoustic coupling between the earpiece and microphone in the handset, what you hear is also sent to the person talking. That person thus not only hears you, but also hears her own speech delayed (because of propagation delay over the telephone network) and attenuated (the acoustic coupling gain is less than one). Furthermore, the same problem applies to you as well: The acoustic coupling occurs in her handset as well as yours.

  1. Develop a block diagram that describes this situation.
  2. Find the transfer function between your voice and what the listener hears.
  3. Each telephone contains a system for reducing echoes using electrical means. What simple system could null the echoes?

No answer provided.
Problem 12:

Demodulating an AM Signal

Let st s t denote the signal that has been amplitude modulated. =xtA1+stsin2πfct x t A 1 s t 2 fc t Radio stations try to restrict the amplitude of the signal st s t so that it is less than one in magnitude. The frequency fc fc is very large compared to the frequency content of the signal. What we are concerned about here is not transmission, but reception.

  1. The so-called coherent demodulator simply multiplies the signal xt x t by a sinusoid having the same frequency as the carrier and lowpass filters the result. Analyze this receiver and show that it works. Assume the lowpass filter is ideal.
  2. One issue in coherent reception is the phase of the sinusoid used by the receiver relative to that used by the transmitter. Assuming that the sinusoid of the receiver has a phase φφ, how does the output depend on φφ? What is the worst possible value for this phase?
  3. The incoherent receiver is more commonly used because of the phase sensitivity problem inherent in coherent reception. Here, the receiver full-wave rectifies the received signal and lowpass filters the result (again ideally). Analyze this receiver. Does its output differ from that of the coherent receiver in a significant way?

No answer provided.
Problem 13:

Jamming

Sid Richardson college decides to set up its own AM radio station KSRR. The resident electrical engineer decides that she can choose any carrier frequency and message bandwidth for the station. A rival college decides to jam its transmissions by transmitting a high-power signal that interferes with radios that try to receive KSRR. The jamming signal jamt jam t is what is known as a sawtooth wave having a period known to KSRR's engineer.

Figure 7
Figure 7 (sig41.png)
  1. Find the spectrum of the jamming signal.
  2. Can KSRR entirely circumvent the attempt to jam it by carefully choosing its carrier frequency and transmission bandwidth? If so, find the station's carrier frequency and transmission bandwidth in terms of TT, the period of the jamming signal; if not, show why not.
No answer provided.
Problem 14:

AM Stereo

A stereophonic signal consists of a "left" signal lt l t and a "right" signal rt r t that conveys sounds coming from an orchestra's left and right sides, respectively. To transmit these two signals simultaneously, the transmitter first forms the sum signal s+t=lt+rt s+ t l t r t and the difference signal s-t=lt-rt s- t l t r t . Then, the transmitter amplitude-modulates the difference signal with a sinusoid having frequency 2W 2 W , where WW is the bandwidth of the left and right signals. The sum signal and the modulated difference signal are added, the sum amplitude-modulated to the radio station's carrier frequency fc fc , and transmitted. Assume the spectra of the left and right signals are as shown.

Figure 8
Figure 8 (spectrum13.png)
  1. What is the expression for the transmitted signal? Sketch its spectrum.
  2. Show the block diagram of a stereo AM receiver that can yield the left and right signals as separate outputs.
  3. What signal would be produced by a conventional coherent AM receiver that expects to receive a standard AM signal conveying a message signal having bandwidth WW?
No answer provided.
Problem 15:

Novel AM Stereo Method

A clever engineer has submitted a patent for a new method for transmitting two signals simultaneously in the same transmission bandwidth as commercial AM radio. As shown, her approach is to modulate the positive portion of the carrier with one signal and the negative portion with a second.

Figure 9
Figure 9 (amstereo.png)

In detail the two message signals m1t m1 t and m2t m2 t are bandlimited to WW Hz and have maximal amplitudes equal to 1. The carrier has a frequency fc fc much greater than WW. The transmitted signal xt x t is given by xt=A1+am1tsin2πfctifsin2πfct≥0A1+am2tsin2πfctifsin2πfct<0 x t A 1 a m1 t 2 fc t 2 fc t 0 A 1 a m2 t 2 fc t 2 fc t 0 In all cases, 0<a<1 0 a 1 . The plot shows the transmitted signal when the messages are sinusoids: m1t=sin2πfmt m1 t 2 fm t and m2t=sin2π2fmt m2 t 2 2 fm t where 2fm<W 2 fm W . You, as the patent examiner, must determine whether the scheme meets its claims and is useful.

  1. What is the receiver for this scheme? It would yield both m1t m1 t and m2t m2 t from xt x t .
  2. Find the spectrum of the positive portion of the transmitted signal.
  3. Determine whether this scheme satisfies the design criteria, allowing you to grant the patent. Explain your reasoning.

No answer provided.
Problem 16:

A Radical Radio Idea

An ELEC 241 student has the bright idea of using a square wave instead of a sinusoid as an AM carrier. The transmitted signal would have the form xt=A1+mtsqTt x t A 1 m t sqT t where the message signal mt m t would be amplitude-limited: |mt|<1 m t 1

  1. Assuming the message signal is lowpass and has a bandwidth of W Hz, what values for the square wave's period T are feasible. In other words, do some combinations of W and T prevent reception?
  2. Assuming reception is possible, can standard radios receive this innovative AM transmission? If so, show how a coherent receiver could demodulate it; if not, show how the coherent receiver's output would be corrupted. Assume that the message bandwidth W=5kHz W 5 kHz .

No answer provided.

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