What is Fourier Series? A Fourier series is an expansion of a periodic function in terms of an infinite sum of sines and cosines. Fourier series make use of the orthogonality relationships of the sine and cosine functions. The computation and study of Fourier series is known as harmonic analysis and is extremely useful as a way to break up an arbitrary periodic function into a set of simple terms that can be plugged in, solved individually, and then recombined to obtain the solution to the original problem or an approximation to it to whatever accuracy is desired or practical. Examples of successive approximations to common functions using Fourier series are illustrated above. Why do We Need Fourier Analysis? Harmonically Related Complex Exponentials (cont’d) • The essence of Fourier analysis is to represent signals in terms of complex exponentials (or trigonometric functions) x(t ) = ∞ ∑ ak e jω 0 t k = −∞ = ... + a− 2e jω0t + a−1e jω 0 t + a0 + a1e jω0t + a2 e jω0t • x(t ) = + ... • Many reasons: – Almost any signal can be represented as a series of complex exponentials – A compact way of approximating several signals. This opens a lot of applications: • storing analog signals (such as music) in digital environment • over a digital network, transmitting digital equivalent of the signal instead of the original analog signal is easier! Thus, a linear combination of them is also periodic!: ∞ ∑C e k = −∞ • • • • k jkω0t So, that means it should be possible to split a periodic signal into a set of periodic signals with same fundamental frequency. This above representation of a periodic signal is referred as Fourier Series representation of that signal. In the Fourier series above, the terms for k=1 and k=-1 are referred as the fundamental components or the first harmonic components of x(t). There are a number of different forms a Fourier Series can take. They are equivalent. Engineers work with the Exponential form. 1 Fourier Series Coefficients Find its Fourier Series coefficients Ck = Example 1 Given a signal y(t) = cos(2t), find its Fourier Series coefficients. Example : Calculate the Fourier Series coefficients for the impulse train. 1 x(t )e − jkω0t dt T T∫ 2 Example : Calculate the Fourier Series of a square wave . Chapter-5 Fourier Transform In the previous three lessons, we discussed the Fourier Series, which is for periodic signals. This lesson will cover the Fourier Transform which can be used to analyze aperiodic signals. (Later on, we'll see how we can also use it for periodic signals.) The Fourier Transform is another method for representing signals and systems in the frequency domain the ω-axis, distance between two consecutive aks is now ω0=2π/T, the fundamental frequency. On 3 Bridge Between Fourier Series and Transform • Consider the periodic signal x(t) below: x(t) -T t -T1 0 T1 T • We know that the Fourier coefficients for x(t) will be:  2T1 ,  T ak =   sin(kω 0T1 )  kπ Bridge Between Fourier Series and Transform (cont’d) • Now, sketch ak on the ω-axis: ak 2T1/T -2ω0 -ω0 0 ω0 2ω0 ω • On the ω-axis, distance between two consecutive aks is now ω0=2π/T, the fundamental frequency. k =0 k ≠0 Bridge Between Fourier Series and Transform (cont’d) • As the period TÆ∝, the fundamental frequency ω0Æ0. So, the distance between the two consecutive aks becomes zero, and the sketch of ak becomes continuous, what is called as Fourier Transform. • At the other side, as TÆ∝, the signal x(t) becomes aperiodic and takes the form: x(t) -T1 0 T1 t • This means the Fourier Transform can represent an aperiodic signal on the frequency-domain. 4 Example Find the Fourier Transform of cos(ω0t) Multiplication of Signals • It states that the Fourier Transform of the product of two signals in time is the convolution of the two Fourier Transforms. 5 Proof Therefore, Fourier Transforms of Sampled Signals In this lesson, we will discuss sampling of continuous time signals. Sampling a continuous time signal is used, for example, in A/D conversion, such as would be done in digitizing music for storage on a CD, digitizing a movie for storage on a DVD, or taking a digital picture. Now, we will derive the Sampling Theorem . To do this, we will examine our signals in the frequency domain. To start, let p(t) have a Fourier Transform P(ω), x(t) have a Fourier Transform X(ω), and xs(t) have a Fourier Transform Xs(ω). Then, because xs(t) = x(t)p(t), by the Multiplication Property, Now let's find the Fourier Transform of p(t). Because the infinite impulse train is periodic, we will use the Fourier Transform of periodic signals: where Ck are the Fourier Series coefficients of the periodic signal. 6 Let's find the Fourier Series coefficients Ck for the periodic impulse train p(t): Therefore Now, to finish our derivation of the Sampling Theorem, we will go back and determine Xs(ω). We saw that: Therefore, or we get replicated, scaled versions of X(ω), spaced every ω0 apart in frequency: From this development and observing the above figure, we can derive our Sampling Theorem. This is one of the most important results. 7 We can recover x(t) from its sampled version xs (t) by using a low pass filter to recover the center island: After low pass filtering As you can see from the figure if ω0 - ωc < ωc, we would get overlap of the replicates of X(ω) in frequency. This is known as "aliasing." Therefore, to avoid aliasing, we require ω0 - ωc > ωc or ω0 > 2ωc. If we avoid aliasing, we can recover x(t) from its samples. (Usually, we choose a sampling rate a bit higher than twice the highest frequency since filters are not ideal.) We hear music up to 20 kHz and CD sampling rate is 44.1 kHz. (Dogs would need a higher quality CD since they hear higher frequencies than humans.) Sampling Theorem Conditions Sampling Theorem: A continuous-time signal x(t) can be uniquely reconstructed from its samples xs(t) with two conditions: – x(t) must be band-limited with a maximum frequency ωM – Sampling frequency ωs of xs(t) must be greater than 2ωM, i.e. ωs>2ωM. • The second condition is also known as Nyquist Criterion. • ωs is referred as Nyquist Frequency, i.e. the smallest possible sampling frequency in order to recover the original analog signal from its samples. Aliasing (Under-sampling) • What happens when sampling frequency is less than Nyquist Frequency, i.e. ωs<2ωM ? • The original signal x(t) cannot be recovered from xs(t) since there will be unwanted overlaps in Xs(ω). • This will cause the recovered signal xr(t) to be different than x(t), also called as aliasing or under-sampling. 8 Example 2 The inverse Fourier Transform of the signal in the previous example is Draw the sampled signals using the sampling trains of the previous example 9 Example 2 AM radio - Double-sideband, suppressed carrier, amplitude modulation • Let x(t) be a music signal with Fourier Transform X(ω). To modulate this signal, we'll form and transmit Your radio demodulates the signal by multiplying the received signal y(t) by another cosine wave to form a third signal z(t): 10 To recover the original music signal x(t) from the demodulated signal z(t), you filter z(t) with a low-pass filter by multiplying by a rect function in frequency. This corresponds to convolving with a sinc function in time. The Bilateral Laplace Transform of a signal x(t) is defined as: The complex variable s = σ + jω, where ω is the frequency variable of the Fourier Transform (simply set σ = 0). The Laplace Transform converges for more functions than the Fourier Transform since it could converge off of the jω axis. Here is a plot of the s-plane: The xy-axis plane, where x-axis is the real axis and y-axis is the imaginary axis, is called as splane. We will perform our visualizations of Laplace transform on s-plane 11 Region of Convergence (ROC) • Similar to the integral in Fourier transform, the integral in Laplace transform may also not converge for some values of s. • So, Laplace transform of a function is always defined by two entities: – Algebraic expression of X(s). – Range of s values where X(s) is valid, i.e. region of convergence (ROC). bilateral and unilateral Laplace Transforms As we'll see, an important difference between the bilateral and unilateral Laplace Transforms is that you need to specify the region of convergence (ROC) for the bilateral case. We point out (without proof) several features of ROCs: • A right-sided time function (i.e. x(t) = 0, t < t0 where t0 is a constant) has an ROC that is a right half-plane. • A left-sided time function has an ROC that is a left half-plane. • A 2-sided time function has an ROC that is either a strip or else the ROC does not exist, which means that the Laplace Transform does not exist. 12 or 13 Example Example Recall the equation for the voltage of an inductor: If we take the Laplace Transform of both sides of this equation, we get: which is consistent with the fact that an inductor has impedance sL 14 Example 1 Find y(t) where the transfer function H(s) and the input x(t) are given. Use Partial Fraction Expansion to find the output y(t): Example 5 Find the output of an LTI system with impulse response h(t) = ebtu(t) to an input x(t) = eatu(t), where a ≠ b. 15 16 17 18 19 20