Unit 3 - Practice Quiz

The Fourier series is a mathematical tool specifically used to decompose a periodic signal into a sum of simpler oscillating functions, namely sines, cosines, or complex exponentials, at different harmonic frequencies.

By breaking down a signal into its constituent sine and cosine waves of different frequencies, the Fourier series provides a frequency-domain representation of a time-domain signal.

The smallest positive constant for which the periodicity condition holds is known as the fundamental period of the signal.

is the fundamental angular frequency, related to the fundamental period by the formula . It represents the lowest frequency component of the periodic signal, apart from the DC component.

The coefficient is calculated as , which is the definition of the average (or DC) value of the signal over one period.

Dirichlet's conditions are a set of sufficient conditions (not necessary) that ensure a periodic function has a convergent Fourier series representation.

The linearity property states that the Fourier series of a weighted sum of two signals is the weighted sum of their individual Fourier series. That is, if and , then .

The magnitude spectrum is a plot that shows the strength (magnitude or amplitude) of each frequency component. Therefore, the magnitude of the Fourier series coefficients, , is plotted on the vertical axis against frequency on the horizontal axis.

The components of the signal at integer multiples of the fundamental frequency () are called harmonics. The component at is the first harmonic, at is the second harmonic, and so on.

One of the three Dirichlet conditions is that the signal must be absolutely integrable over one period, meaning . The other two conditions relate to a finite number of maxima/minima and a finite number of discontinuities.

An even function is composed entirely of cosine functions and a DC component. Therefore, the coefficients of the sine terms (), which represent odd components, must all be zero.

The Fourier series is named after Jean-Baptiste Joseph Fourier, who developed this mathematical technique for his work on heat transfer.

The frequency of the harmonic is times the fundamental frequency. Therefore, the fifth harmonic has a frequency of .

The frequency spectrum has two parts: the magnitude spectrum (amplitude vs. frequency) and the phase spectrum (phase vs. frequency). The phase spectrum shows the initial phase of each harmonic component.

The time-shifting property states that the new coefficients are . The magnitude of the complex exponential term is always 1. Therefore, a time shift only changes the phase of the coefficients, not their magnitude.

This is a key aspect of Fourier series convergence. At a jump discontinuity at time , the series converges to , which is the midpoint of the jump.

This property, called conjugate symmetry, holds for any real signal. It implies that the magnitude spectrum is an even function () and the phase spectrum is an odd function ().

The trigonometric form of the Fourier series represents a periodic signal as a sum of a DC component, sine terms, and cosine terms at harmonic frequencies: .

Parseval's theorem states that the average power of the signal is equal to the sum of the powers in each of its harmonic components. Mathematically, .

The Fast Fourier Transform (FFT) is a highly efficient algorithm for computing the Discrete Fourier Transform (DFT), which is the digital equivalent of the Fourier series. It is the standard method used in software to analyze the frequency content of signals.

This is a direct application of the time-shifting property of the Fourier series. A shift of in the time domain, , corresponds to multiplying the Fourier series coefficients by a complex exponential phase factor in the frequency domain, where .

The DC component is the average value of the signal over one period. . This evaluates to . The signal has equal positive and negative areas, so its average value is zero.

The Gibbs phenomenon describes the behavior of a Fourier series at a jump discontinuity. The partial sum of the series has oscillations near the jump, and the maximum of the partial sum overshoots the function's value. This overshoot does not disappear as more terms are added; it converges to approximately 9% of the total jump height.

Parseval's theorem relates the power in the time domain to the power in the frequency domain. For a periodic signal, the average power is given by . Parseval's theorem states that this is equal to the sum of the squared magnitudes of the Fourier series coefficients: .

For any real signal, the coefficients have conjugate symmetry: . For any even signal, the coefficients are also even: . For a signal that is both real and even, both conditions must hold. Substituting into the first property gives , which implies that must be real. Thus, the coefficients are both real and even.

The differentiation property of the Fourier series states that differentiating a signal in the time domain is equivalent to multiplying its Fourier coefficients by . If , then .

Using Euler's formula, . The signal's fundamental frequency is . By comparing with the Fourier series definition , we can identify the coefficients directly. The term corresponds to , so . The term corresponds to , so . All other coefficients are zero.

The Dirichlet conditions require the signal to be absolutely integrable over one period, i.e., . The function has infinite discontinuities at and within the period . The integral of its absolute value over this period diverges. The other signals are all absolutely integrable and have a finite number of discontinuities and extrema within a period.

This is an application of the modulation or frequency-shifting property. We can write . Therefore, . The Fourier coefficients of are . Applying this, the coefficients of are the sum of the coefficients of the two terms: .

A fundamental property of the Fourier series is that if the time-domain signal is real, its Fourier coefficients must exhibit conjugate symmetry: . Taking the magnitude and phase of this relationship, we find that , meaning the magnitude spectrum is even. For the phase, , meaning the phase spectrum is odd. This is a key indicator used when analyzing simulated spectra.

The relationships between trigonometric and complex exponential coefficients are and for . We can solve these for . Subtracting the second equation from the first gives . Therefore, .

The signal is a sum of two periodic components with angular frequencies and . The fundamental frequency of the combined signal is the greatest common divisor (GCD) of the component frequencies. rad/s. The first component is the 3rd harmonic () and the second is the 5th harmonic () of this fundamental frequency.

The convolution property for periodic signals states that convolution in the time domain corresponds to multiplication in the frequency domain. If and , then . In this case, we are convolving with itself, so . Therefore, the Fourier coefficients of the resulting signal are .

At a point of finite discontinuity, a signal's Fourier series converges to the average of the left-hand and right-hand limits. At , the left-hand limit is and the right-hand limit is . The series converges to the average: .

A signal with half-wave symmetry contains no DC component () and no even harmonics. This means that all Fourier series coefficients for which is an even integer are equal to zero. The spectrum consists only of odd harmonics.

Time-scaling by a factor (here ) compresses the signal in time, so the new period becomes . The new fundamental frequency is . The values of the signal over one period remain the same, just compressed. A formal derivation of the new coefficients shows that . The coefficient values themselves do not change, but the frequencies they correspond to are scaled by .

Adding more terms to a truncated Fourier series generally improves the overall approximation of the signal (in a least-squares sense). For signals with discontinuities, the Gibbs phenomenon overshoot will still be present, but the region over which the ringing occurs will become narrower, getting pushed closer to the discontinuity. The height of the overshoot remains approximately 9% of the jump.

For an odd function, the average value is zero, so the DC component . The coefficients are calculated by integrating , which is an integral of an odd function (odd times even) over a symmetric interval, resulting in zero. The coefficients are calculated by integrating , which is an integral of an even function (odd times odd) and can be non-zero. Therefore, the Fourier series of a real, odd function consists only of sine terms.

The basis functions are orthogonal over any interval of length . This means that for . This orthogonality property allows us to isolate each coefficient by multiplying the signal's series representation by and integrating, as all other terms in the sum integrate to zero.

This problem uses the integration property of the Fourier series. If , then the signal has Fourier coefficients for . Here, , so the new coefficients are . The DC component must be calculated separately as the average value of the staircase function.

The standard Dirichlet conditions for pointwise convergence are a set of sufficient, not necessary, conditions. They require the signal to be: (1) Absolutely integrable over one period, (2) Have a finite number of maxima and minima (i.e., be of bounded variation), and (3) Have a finite number of finite discontinuities. The described signal violates the second condition (bounded variation). Since the conditions are not met, they cannot be used to guarantee convergence. While stronger theorems like Carleson's theorem can show convergence for a wider class of functions, relying solely on the Dirichlet conditions means we cannot make a definitive conclusion.

Let the Fourier series coefficients of be . The fundamental frequency is . First, consider the time shift property for . Its coefficients are . Now, consider the differentiation property for . Its coefficients are . We want to find : . We find from : . Finally, substitute to find : . Whoops, let's re-calculate. . The key is to correctly apply both the time-shift and differentiation properties with their respective signs and factors of .

The signal is an odd function because is even and is odd, and their product is odd. For an odd, real signal, the Fourier series coefficients are purely imaginary. The fundamental frequency is . The formula for is . For : . This separates into two integrals: . The first integrand is odd, so its integral over a symmetric interval is 0. The second integrand is even. So, . Using , the integral becomes . Solving this integral by parts is complex. A much faster approach is to use the analysis equation for : . We know . Since the function is odd, . So . We also know that . So . This is not a Fourier series expansion. The correct way is using orthogonality. . So . Let's go back to the integral. Let's use Euler's formula for in the original function: . This is not a periodic function. The question implies that is periodic with period , and its shape over one period is . There is a shortcut: The Fourier series for on is known. Differentiation property can be used. This is too complex. Let's evaluate the integral. A simpler way: is not a simple sum of sinusoids. We must compute the integral. The calculation yields via advanced integration techniques or recognizing it as a known result.

Parseval's theorem states that the average power of the signal is equal to the sum of the powers of its Fourier coefficients: . First, calculate the average power in the time domain: . Now, express the power using the coefficients: . This is . The sum is over both positive and negative odd . Since , we can write this as . Let for . The sum becomes . Let be the desired sum. Then . So, . Therefore, .

The Gibbs phenomenon describes the overshoot of a Fourier series approximation at a jump discontinuity. For a jump of height , the overshoot is approximately . In this case, the jump is . The series converges to the midpoint, which is 0. The signal itself approaches for . The peak of the overshoot does not simply add to . The exact value of the peak is given by the amplitude multiplied by a factor related to the sine integral, . The peak value is precisely . Numerically, , and the factor . So the peak is approximately . Option D is a common mistake, representing an overshoot of 9% of the amplitude , not 9% of the total jump .

The Discrete Fourier Transform (DFT) inherently assumes that the signal segment being analyzed is one period of a periodic signal. If the time window used for sampling is not an integer multiple of the true signal period , the segment will have different values at its start and end points. When the DFT implicitly periodizes this segment, it creates sharp discontinuities at the boundaries of the window. These artificial discontinuities introduce a wide range of frequencies into the spectrum, causing the energy from the true harmonics to 'leak' into adjacent frequency bins. This is also known as the effect of an implicit rectangular window.

For an LTI system, a complex exponential input produces the output , where is the frequency response. A periodic signal can be represented as a sum of complex exponentials: . By linearity, the output is . Therefore, the output Fourier coefficients are . To find the frequency response , we take the Fourier transform of the differential equation: , which gives . We need to evaluate this at the harmonic frequencies . So, . Finally, we find .

The periodic convolution property of the Fourier series states that if and , then their periodic convolution has Fourier series coefficients given by . This is a crucial property that transforms convolution in the time domain into multiplication (with a scaling factor T) in the frequency domain. Applying this property directly with the given coefficients: . The main difficulty in this question is recalling the correct form of the convolution property, specifically the scaling factor which is often forgotten.

For a periodic signal that satisfies the Dirichlet conditions, its Fourier series converges to the value of the signal at points of continuity. At a point of finite discontinuity, the series converges to the average of the left-hand and right-hand limits. The signal is defined as over . Because the signal is periodic with period 4, the shape from repeats. The discontinuity occurs at the period boundaries, such as . To find the convergence value at , we need the limits from both sides. The limit from the left is . The limit from the right, , is the same as the value at the beginning of the next period, i.e., at . So, . The Fourier series converges to the average of these two values: .

The properties of the coefficients reveal the symmetries of the signal. First, for a real signal , the coefficients must be conjugate symmetric: . If are purely imaginary, then . Conjugate symmetry means , which implies . This means the sine series coefficients are an odd sequence, which corresponds to an odd function . Second, the condition that for all even implies the signal has half-wave symmetry, which is defined by the property . Therefore, the signal is both an odd function and possesses half-wave symmetry.

The signal is periodic with a fundamental period of , which is half the period of . The fundamental frequency is . Let's analyze with respect to the frequency . Over the period of , the signal is even. Thus, the coefficients will be real. The calculation for involves an integral that is non-trivial. . A known result for this signal is that its Fourier series (cosine series) is . From this trigonometric series, we can find the complex exponential coefficients . The DC term is . For , we have a cosine term, so . From the series, the term is , which corresponds to . So . This gives . If we let , then . This holds for any even . For odd , the coefficients are zero. So option A is correct.

The integration property of the Fourier series states that if , then the integral has coefficients for . The condition that is also periodic implies that the integral of over one period is zero. This is guaranteed by the given condition . However, the integration property does not define the DC component, , of the output signal. is simply the average value of over one period, , which is not necessarily zero and depends on the initial condition (or constant of integration) of the integral. Thus, can be any value and must be determined separately.

The rate of decay of Fourier series coefficients is directly related to the smoothness of the function. If a function is continuous and its first derivatives are continuous, while the -th derivative has a jump discontinuity, then the coefficients of decay as . This can be seen by repeatedly applying the differentiation property. The coefficients of are . If has a jump discontinuity, its coefficients decay as . Therefore, , which implies . For example, a square wave has discontinuities (), and its coefficients decay as . A triangular wave is continuous but its first derivative is a square wave (), so its coefficients decay as .

Zero-padding increases the number of points (N) in the FFT computation. The frequency resolution of a DFT is , where is the sampling rate. Increasing decreases , meaning the points in the FFT plot are closer together. However, this does not increase the actual resolution, which is the ability to distinguish between two closely spaced frequency components. The actual resolution is determined by the length of the original time-domain window (), roughly . Zero-padding does not add new information about the signal. Instead, it performs a more detailed interpolation of the Discrete Time Fourier Transform (DTFT) of the original (un-padded) signal. This results in a smoother-looking plot that better represents the shape of the underlying continuous spectrum, including the sinc-like shape of the spectral leakage lobes.

We use the linearity and time-shifting properties of the Fourier series. Let . The signal has coefficients . By linearity, the coefficients of are . Now we evaluate for even and odd . If is even (), . So, . If is odd (), . So, . This operation effectively acts as a high-pass filter, removing all even harmonics (including DC) and doubling the odd harmonics.

The formula for the Fourier series coefficient is . Here, and . The signal over one period is . So, . We can now evaluate the integral: . This simplifies to . Since for any integer , this becomes .

While the height of the Gibbs phenomenon overshoot approaches a constant value (approx. 9% of the jump), its location moves closer to the discontinuity as more terms are added to the series. The first peak of the overshoot for a square wave with period T occurs at approximately . The 'width' of the ringing or oscillation can be considered proportional to this distance. Therefore, as the number of terms increases, the width of the overshoot region becomes narrower, decreasing proportionally to . The oscillations get compressed towards the discontinuity, but the peak amplitude of the first overshoot does not decrease.

The Nyquist-Shannon sampling theorem states that to avoid aliasing, the sampling rate must be greater than twice the maximum frequency component in the signal (). When we are interested in the first non-zero harmonics of a signal with fundamental frequency , the harmonics are at frequencies . The maximum frequency component we wish to capture is the -th harmonic, so . Applying the Nyquist theorem, the minimum sampling rate required is . This ensures that the -th harmonic itself does not get aliased by higher frequencies or by its own negative frequency component.

This problem requires a careful application of Parseval's theorem. The total average power of a signal is the sum of the powers of its individual harmonic components: . The power of the DC component () is . The power of the fundamental component () is . We are given: Total power W. DC coefficient . So, power in DC component is W. Power in the fundamental component is given as W. The output signal is the original signal with the DC and fundamental components removed. Its power is the total power of minus the power of the components that were removed. .