Fourier Series & Fourier Transform

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Response of LTI- Systems to complex exponentials and periodic signalsConsider an exponential signal, x1(t) = exp(j0t) = exp(s0t) is applied at the input of an LTI-system having an impulse response, h(t), where 0 is fundamental frequency of x1(t). We are interested in the output, z(t) of this system. We already know that it is simply the analog convolution of x1(t) with h(t) and we compute it mathematically as

We conclude that the output, z(t) is simply the product of a complex constant, i.e., (Eigen value computed at the frequency of the input signal/component and the input, x1(t) which contains only one frequency component. Now, we introduce harmonically related frequency components in the input signal, x1(t) by multiplying its fundamental frequency, 0 with the integer, k and apply it at the input of the same system having h(t) as its impulse response. Thus, we express the output z1(t) generated by this system using the above result as

Now, we want to compute the output, y(t) of the system due to the input, x(t), i.e., this representation of x(t) is well known as Fourier Series, . We can easily compute the output, y(t) of the system by exploiting LTI properties of the system as

To summarize the above process, kindly refer to the following illustration

Determination of the output due to periodic signals, Acos(0t) and Asin(0t)

How to find out the Fourier Series Coefficients, ak ?

Procedure of computing aks

1. x(t) is a given real periodic signal, determine its fundamental period, T0 and fundamental frequency, 02. Multiply x(t) with exp(-jk0t) = cos(k0t) jsin(k0t)3. Integrate step # 02 w.r.t. t over one fundamental period of x(t); Choice / selection of T0 is quite important in reducing the effort of computing the above integral4. If x(t) is even function of t, then aks are pure real; on the other hand, if x(t) is an odd function of t, then aks are pure imaginary5. Multiply the result of step # 04 by (T0)-1. This will yield the Fourier series coefficient, ak.

Representation of x(t) into Fourier series and construction of its spectrum1. Compute Fundamental frequency, 0 from the relation 0 = 2/T0, where T0 represents the fundamental period of x(t)2. Compute the Fourier series coefficients, aks using the above result and illustrated procedure3. Usually the Fourier series coefficients are complex in nature. Thus, its spectrum comprises of two parts; (1) Magnitude spectrum, |ak| and (2) the Phase spectrum, arg(ak)4. For the plot of ak, we usually take (k0) along x-axis by varying k and either the magnitude or phase of ak along y-axis. Hence, we realize that spectrum consists of distinct lines and therefore it is known as line spectrum and it exhibits Hermitian symmetry. As x(t) is a real periodic signal, therefore, |ak| is an even function of frequency and arg(ak) is an odd function of frequency as demonstrated below:

Conversion of Exponential Fourier Series into Trigonometric Fourier Series

Filtering of x(t) expressed in terms of Fourier Series, i.e., Determination of bks

1. Express x(t) in Fourier series representation. It means that we already know aks2. Compute the frequency response of the filter using its impulse response, h(t) using the following relation; 3. Evaluate the frequency response at = k0 to yield H(jk0)4. Formulate the product of ak with H(jk0) which is equal to bks and it is also complex5. Moreover, (bk)* = (ak)*H*(jk0) = a-k H(-jk0) = b-k. This shows that bks also exhibit Hermitian symmetry. Lowpass and highpass Filters of a first order RC NetworkConsider an input signal, x(t) is applied across a series combination of RC network. An output signal, y(t) is measured across C. Let us first develop input-output relation of this network as mentioned below:

Properties of Fourier Series Coefficients, akWe extensively utilize these properties to reduce the amount of effort in determining the Fourier series coefficients of the periodic signal. Thus, knowledge of these properties is utmost important and the way how to apply them should also be very clear and understandable. We highlight these properties as;

Thus, we conclude that time scaling effect on x(t) does not bring about any change in the result of Fourier series coefficients. Time scaling does change the fundamental frequency, 0 of the periodic signal, x(t) and produces a similar opposite impact on the fundamental period, T0 of x(t). That is why, the value of Fourier series coefficients, aks remains unaltered.

Extension to Discrete-time periodic signalsContinuous-time Periodic Signal, x(t)Discrete-time Periodic Signal, x[n]

Fundamental frequency0 = 2 / T0 ; T0 represent fundamental period of x(t); w0 = 2 / N0 ; N0 represent fundamental period; - < w0 <

Harmonic components

Infinite ; k0 = 2k / To ; Finite (N) ; kw0 = 2k / No ;

Input-output Relation

Fourier series representation

Fourier series coefficient, ak

PropertiesTime Scalingxm[n] = x[n/m] ; if n = rm and zero otherwise; ak / m ; viewed as periodic with period mN0.

Multiplicationz[n] = x[n]y[n]; ck = ak * bk over one period, N0

DifferentiationFirst Difference; x[n] x[n 1](1 exp(-j2/N0)) ak

IntegrationRunning sum; finite valued and periodic only if a0 = 0

Parsevals relation

Filtering of the input periodic discrete-time signal, x[n] To summarize the above process, kindly refer to the following illustration

It is important to note here that both magnitude and phase plots of Discrete-time filter for different values of a are periodic with period 2 in contrast to frequency responses of continuous-time filters which are aperiodic. The frequency response of the discrete-time filter is a continuous function of frequency, w.

Continuous-time Fourier Transform (CTFT)The objectives of this chapter are illustrated below: Illustrate students the importance of the frequency domain Differentiate between Fourier Series and Fourier Transform and their respective spectra Illustrate the usefulness of the properties of Fourier Transform Illustrate how convolution can be handled in frequency domain with ease and comfort Illustrate how the input-output relation (differential equation) and the output generated by the system, y(t) can easily be solved/computed in frequency domain Illustrate how ak can be computed from the knowledge of X(j) How to draw the frequency response of the system using its propertiesWe already know that spectrum of continuous time Fourier series coefficients, ak comprises of distinct lines separated by 0 and is known as line spectrum. But as fundamental period, T0 of x(t) is increased, its fundamental frequency, 0 starts decreasing and in the limit when T0 becomes infinite (very large), then 0 becomes ideally zero (very small) which implies that the periodic signal, x(t) does remain no longer periodic and its line spectrum is converted into continuous spectrum. The aperiodic signal, x(t) may be recovered as

Properties of CTFT

PropertyDefinition in time domainEnd result is frequency domain

Linearity

Time Reversal

Time Shifting

Time Scaling

Conjugation

Duality

Differentiation

Integration

Convolution

Correlation

Parsevals Relation

Periodic Signals

Even and odd real signals

Shifting, Differentiation, Integration and Convolution in frequency DomainIllustrate these properties with the help of Duality Property as discussed above.

Instructor is advised to illustrate the usefulness of each property by doing an example so that students should be able to see what is happening in frequency and time domains.Example 1: Determine the Fourier Transform of the signal, x(t) = Aexp(-Rt)u(t) using its definition

Example 2: Determine the Fourier Transform of Unit Step Function, u(t) using the result of example 1.

The variation of the first term on the RHS w.r.t to R should be explained to the students. Its value at = 0 is 1/R and as we decrease the value of R, 1/R gets increased and its rate of decay becomes faster (width decreases). The total area bounded by this function over the entire -axis is simply equal to and in the limit, it gets the shape of an impulse on -axis. Thus,

Example 4: Now, well make use of example 2 in explaining the properties of CTFT and well be able to compute the Fourier transforms of Rectangular and Triangular pulses.

Computation of Frequency response, Impulse Response and Step response from the given Differential Equation (input-output Relation) The following steps are essential in finding the answers of the above questions:1. Compute the Fourier transform of the given differential (integro-differential) Equation using differentiation and integration properties of Fourier transforms. This will transform the given input-output relation (diff. equation) into Algebraic equation.2. Solve this equation for the ratio Y(j) / X(j) which is equal to H(j), the frequency response of the system. 3. This frequency response is generally a complex quantity. Determine its magnitude and phase using either Algebra of Complex Numbers (or Mathematical review of Chapter 1).4. Sketch its spectra by taking frequency, on x-axis and magnitude or phase on y-axis. This will provide us the magnitude and phase plots of the given system5. All those values of which result in H(j) = 0 will provide us zeros of the systems (the system does not allow these frequencies to pass through it). 6. All those values of which result in H(j) = infinity will provide us poles of the systems (the system usually is not allowed to operate on these frequencies). 7. If all poles of the system are negative (lie in the left half s-plane, then the system is stable otherwise not). Zeros dont play any role in determining the stability of the system.8. Use IFT techniques (usually partial fractions and tables) to determine the impulse response of the system, i.e., h(t) = IFT{H(j)}9. Step response, s(t) of the system is usually determined by integrating its impulse response10. Having determined the frequency response of the system, the output, y(t) due to any input x(t) can easily be computed by performing the following two steps: Multiply H(j) with X(j). This is usually equal to Y(j) Use IFT techniques to compute y(t) Illustration with the help of an example

Chapter 5:Discrete-time Fourier Transform (DTFT) Illustrate the students the importance of the frequency domain and the difference between the analysis and synthesis equations of both transforms Differentiate between Discrete-time Fourier Series and Discrete-time Fourier Transform and their respective spectra with that of CT Fourier series and FT Illustrate the usefulness of the properties of Discrete-time Fourier Transform Illustrate how convolution can be handled in frequency domain with ease and comfort Illustrate how the input-output relation (difference equation) and the output generated by the system, y[n] can easily be solved/computed in frequency domain Illustrate how ck can be computed from the knowledge of X(ejw) How to draw the frequency response of the system using its propertiesContinuous DomainDiscrete Domain

Fourier Series Representation

Fourier Series Coefficients, ak

SpectrumDiscrete (Line Spectrum); AperiodicDiscrete (Line Spectrum); Periodic with period = 2

Fourier Transform (Analysis Equation)

Fourier Transform (Synthesis Equation)

Relation of ak with X(j)

SpectrumContinuous and AperiodicContinuous and Periodic with period = 2

Frequency ResponseH(j) = Y(j) / X(j); AperiodicH(ejw) = Y(ejw) / X(ejw); Periodic

ApplicationUsed to solve the differential equationUsed to solve the difference equation

Step Response

Causality & Stabilityh(t) = 0 for t < 0 and the poles must have negative real parts.h[n] = 0 for n < 0 and the magnitudes of all poles must be less than unity.

PropertiesBoth have the same properties except time scaling and duality in case of DT signals

Examples: Compute Discrete-time Fourier transforms of some elementary signals and sketch their specra

Properties of DTFT and its comparison with CTFT

PropertyCTFTDTFT

Analysis Equation

Synthesis Equation

Linearity

Time Reversal

Time Shifting

Time Scaling

Conjugation

Duality

No such relation exists in this case

Differentiation / Differencing in time

Integration / Accumulation

Convolution

Correlation

Parsevals Relation

Periodic Signals

Even and odd real signals

Shifting, Differentiation, Integration and Convolution in frequency DomainIllustrate these properties with the help of Duality Property as discussed above.