Chapter 3, Fourier Transform Video Solutions, Signals and Systems for Bioengineers, Second Edition: A MATLAB-Based Introduction (Biomedical Engineering)

02:14

Problem 1

Find the Fourier series of the square wave below using noncomplex analytical methods (i.e., (3.11) and (3.12)).

James Kiss

Numerade Educator

Problem 2

Find the Fourier series of the waveform below using noncomplex analytical methods. The period, $T$, is $1 \mathrm{sec}$.

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Problem 3

Find the Fourier series of the "half-wave" rectified sinusoidal waveform below using noncomplex analytical methods. Take advantage of symmetry properties.

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Problem 4

Find the Fourier series of the "sawtooth" waveform below using noncomplex analytical methods.

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02:03

Find the Fourier series of the periodic exponential waveform shown below where $x(t)=\mathrm{e}^{-2 t}$ for $0<t \leq 2$ (i.e., $T=2 \mathrm{sec}$ ). Use the complex form of the Fourier series in Equation 3.23.

Kajal Gautam

Numerade Educator

03:12

Problem 6

Find the continuous Fourier transform of an aperiodic pulse signal given in Example 3.5 using the noncomplex equation, Equation 3.28 and symmetry considerations.

Amit Srivastava

Numerade Educator

01:59

Problem 7

Find the continuous Fourier transform of the aperiodic signal shown below using the complex equation, Equation 3.27 .

Amit Srivastava

Numerade Educator

01:59

Problem 8

Find the continuous Fourier transform of the aperiodic signal shown below. This is easier to solve using the noncomplex form, (3.13) and (3.14), in conjunction with symmetry considerations.

Amit Srivastava

Numerade Educator

06:34

Problem 9

Use the MATLAB Fourier transform routine to find the spectrum of a waveform consisting of two sinusoids at 200 and $400-\mathrm{Hz}$. First generate the waveform in a 512-point array assuming a sampling frequency of $1 \mathrm{kHz}$. Take the Fourier transform of the waveform and plot the magnitude of the full spectrum (i.e., 512 points). Generate a frequency vector as in Example 3.10 so the spectrum plot has a properly scaled horizontal axis. Since the DC term will be zero, there is no need to plot it.

Kajal Gautam

Numerade Educator

Problem 10

Use the routine sig_noise to generate a waveform containing 200- and 400-Hz sine waves as in Problem 9, but add noise so that the signal-to-noise ratio (SNR) is $-8 \mathrm{~dB}$; i.e., $\mathrm{x}=$ sig_noise $([200400],-8, N)$ where $\mathrm{N}=512$.Plot the magnitude spectrum, but only plot the nonredundant points $(2$ to $\mathrm{N} / 2)$ and do not plot the DC term, which again is zero. Repeat for an SNR of $-16 \mathrm{~dB}$. Note that the two sinusoids are hard to distinguish at the higher $(-16 \mathrm{~dB})$ noise level.
e points.

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Problem 11

Use the routine sig_noise to generate a waveform containing 200- and $400-\mathrm{Hz}$ sine waves as in Problems 9 and 10 with an SNR of $-12 \mathrm{~dB}$ with 1000 points. Plot only the nonredundant points (no DC term) in the magnitude spectrum. Repeat for the same SNR, but for a signal with only 100 points. Note that the two sinusoids are hard to distinguish with the smaller data sample. Taken together, Problems 10 and 11 indicate that both data length and noise level are important when detecting sinusoids (also known as narrowband signals) in noise.

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Problem 12

Generate the signal shown in Problem 2 and use MATLAB to find both the magnitude and phase spectrum of this signal. Assume that the period, $T$, is $1 \mathrm{sec}$ and assume a sampling frequency of $500-\mathrm{Hz}$; hence you will need to use 500 points to generate the signal. Plot the time domain signal, $x(t)$, then calculate and plot the spectrum. The spectrum of this curve falls off rapidly, so plot only the first 20 points plus the DC term and plot them as discrete points, not as a line plot. Note that every other frequency point is zero, as predicted by the symmetry of this signal.

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Problem 13

Repeat Problem 12 using the signal in Problem 4, but generate the signal between $-T / 2$ and $+T / 2$ (i.e., $\mathrm{x}=\mathrm{t}$; for $\mathrm{t}=(-\mathrm{N} / 2: \mathrm{N} / 2)$; ). Also plot the phase in degrees, that is, scale by $360 /(2 \pi)$. Again plot only the first 20 values (plus the $\mathrm{DC}$ term) as discret

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Problem 14

Repeat Problem 13 for the signal in Problem 4, but generate the signal between 0 and $T$. Unwrap the phase specturm, plot in degrees and plot only the first 20 points (plus DC term) as in Problem 13. Note the difference in phase, but not magnitude, from the plots of Problem 13.

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03:56

Problem 15

Find the discrete Fourier transform for the signal shown below using the manual approach of Example 3.9. The dashed lines indicate one period.

Kajal Gautam

Numerade Educator

03:56

Problem 16

16. Find the discrete Fourier transform for the signal shown below using the manual approach of Example 3.9. The dashed lines indicate one period.

Kajal Gautam

Numerade Educator

06:34

Problem 17

Plot the magnitude and phase components of the ECG signal found as variable ecg in file ECG.mat. Plot only the nonredundant points and do not plot the DC component (i.e., the first point in the Fourier series). Also plot the time function and correctly label and scale the time and frequency axes. The sampling frequency was $125-\mathrm{Hz}$. Based on the magnitude plot, what is the bandwidth (i.e., range of frequencies) of the ECG signal?

Kajal Gautam

Numerade Educator

Problem 18

The data file pulses.mat contains 3 signals: $x 1, x 2$, and $x 3$. These signals are all $1 \mathrm{sec}$ in length and were sampled at $500-\mathrm{Hz}$. Plot the 3 signals and show that each contains a single $40-\mathrm{msec}$ pulse, but at 3 different delays: 0 , 100 , and $200 \mathrm{msec}$. Calculate and plot the spectra for the 3 signals superimposed on a single magnitude and single phase plot. Plot only the first 20 points as discrete points plus the DC term using a different color for each signal's spectra. Apply the unwrap routine to the phase data and plot in degrees. Note that the 3 magnitude plots are identical, while the phase plots are all straight lines but with radically different slopes.

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Problem 19

Load the file chirp.mat which contains a sinusoidal signal, $\mathbf{x}$, that increases its frequency linearly over time. The sampling frequency of this signal is $5000-\mathrm{Hz}$. This type of signal is called a chirp signal because of the sound it makes when played through an audio system. If you have an audio system, you can listen to this signal after loading the file using the MATLAB command: $\operatorname{sound}(x, 5000)$;. Take the Fourier transform of this signal and plot magnitude and phase (no DC term). Note that the magnitude spectrum shows the range of frequencies that are present but there is no information on the timing of those frequencies. Actually, information on signal timing is contained in the phase plot but, as you can see, this plot is not easy to interpret. Advanced signal processing methods known as time-frequency methods are necessary to recover the timing information.

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02:47

Problem 20

Load the file ECG_1min.mat that contains 1 minute of ECG data in variable ecg. Take the Fourier transform. Plot both the magnitude and phase (unwrapped) spectrum up to $20-\mathrm{Hz}$ and do not include the DC term. Find the average heart rate using the strategy found in Example 3.13. The sample frequency is $250-\mathrm{Hz}$.

Vikash Ranjan

Numerade Educator

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