Fourier Analysis of Continuous-Time Signals: Periodic and Aperiodic Signals, Assignments of Biology

Download Fourier Analysis of Continuous-Time Signals: Periodic and Aperiodic Signals and more Assignments Biology in PDF only on Docsity! 051:080 Bioelectrical Design Revised: 01/29/2005 O. Poroy FOURIER ANALYSIS OF CONTINUOUS-TIME SIGNALS PERIODIC, CONTINUOUS-TIME SIGNALS AND THE FOURIER SERIES Topics: The convolution integral, eigenfunctions and eigenvalues, complex exponentials as eigenfunctions of linear, time-invariant systems, the synthesis equation and the analysis equation Fourier Analysis: The techniques of Fourier Analysis form a basis for signal processing methods such as sampling, filtering and modulation. Fourier Analysis can be divided into four groups depending on the signal (and the system) to be analyzed (Figure 6.1). Periodic signals are analyzed using the Fourier Series and aperiodic signals are analyzed using the Fourier Transform. The analysis of continuous-time signals and discrete-time signals is distinguished by inserting the same terms before the word “Fourier.” (Continuous-time signals are signals defined for a continuous range of the independent variable (Example of independent variable: time t). Discrete- time signals are signals defined for discrete values of the independent variable.) The Convolution Integral: The output of a Linear, Time-Invariant (LTI) system can be calculated in the time-domain using the convolution integral. Consider the block diagram for a general system shown in Figure 6.2. The input signal to the system is represented by x(t), and the output of the system is represented by y(t). If the input to the system is a unit impulse function (δ(t), Figure 6.3), the output of the system is called the impulse response. If x(t) = δ(t), then y(t) = h(t). The impulse response, denoted by h(t) is a convenient way to characterize a system in the time-domain. Continuous-time, periodic signals (Continuous-time) Fourier Series Continuous-time, aperiodic signals (Continuous-time) Fourier Transform Discrete-time, periodic signals (Discrete-time) Fourier Series Discrete-time, aperiodic signals (Discrete-time) Fourier Transform Figure 6.1 Classifications of Fourier Analysis )(lim)( 0 tt ∆= →∆ δ ∆ t ∆(t) 1/∆ Figure 6.3 Definition of the Unit Impulse Function System x(t) y(t) Figure 6.2 Block Diagram Representation of a System 051:080 Bioelectrical Design Revised: 01/29/2005 O. Poroy In a time-invariant system, if the input is the unit impulse response shifted in time, the output is the impulse response shifted in time by the same amount: If x(t) = δ(t-τ), then y(t) = h(t-τ). (for a time-invariant system) If the system is also linear, an input of a weighted sum of shifted unit impulse functions results in an output of the same weighted sum of shifted impulse responses: If ∑ +∞ −∞= −⋅= k kk tatx )()( τδ , then ∑ +∞ −∞= −⋅= k kk thaty )()( τ . (for an LTI system) If we take the limit of these equations as τ approaches zero, we obtain the convolution integral: If ∫ +∞ ∞− −⋅= ττδτ dtxtx )()()( , then ∫ +∞ ∞− −⋅= τττ dthxty )()()( . The convolution integral is used to calculate in the time-domain the output signal y(t), for a given input signal x(t), of LTI systems described by an impulse response h(t). Eigenfunctions and Eigenvalues: Consider the system shown in Figure 6.2. If there exists a function, such that when it is input to the system, the output is the same function except for a scaling factor, then that function is called an eigenfunction of the system. The corresponding scaling factor is called an eigenvalue. That is: If )()( tftx e= => )()( tfHty ee ⋅= , then fe(t) is an eigenfunction of the system, and He(s) is the corresponding eigenvalue. Complex Exponentials as Eigenfunctions of LTI Systems: Let the system shown in Figure 6.2 be a linear, time-invariant system, and let stetx =)( , where s is an arbitrary complex number. We can calculate the output y(t) of the system using the convolution integral: ∫∫∫ +∞ ∞− − +∞ ∞− +∞ ∞− ⋅=−⋅=−⋅= ττττττττ τ dehdtxhdthxty ts )()()()()()()( Substituting ττ sstts eee −− ⋅=)( : (Eq’n 6.1) ∫ +∞ ∞− −⋅⋅= ττ τ dehety sst )()( )()()()( txsHesHty st ⋅=⋅= , where ∫ +∞ ∞− −⋅= ττ τ dehsH s)()( 051:080 Bioelectrical Design Revised: 01/29/2005 O. Poroy Example 6.1: Find the Fourier Series coefficients for x(t) = sin ω0t. This function is already expressed as a sum of complex exponentials in Eq’n 6.4. Note that only two exponentials are enough to represent this function. Inspecting Eq’n 6.4: a1 = 1/2j a-1 = -1/2j ak = 0 for all k except k = +/-1 Homework 4: Find the Fourier Series coefficients for the following function x(t): ( ) ( ) ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ +⋅+⋅++= 4 2coscos2sin1)( 000 πωωω ttttx Hint: First term: In order for a complex exponential to have a constant value, k must be zero. Hence, this dc term will determine the coefficient a0. Second and third terms: Follow Example 6.1. Last term: Use Eq’n 6.3 to express it in terms of complex exponentials. Then, use Eq’n 6.1 to separate the exponential with the exponent π/4. This is a constant term (not a function of t) and you can evaluate it using Euler’s Formula. After writing each term as complex exponentials, combine similar terms to find the coefficients. Answer: ( ) ( ) 2,0 22 1 2 1 22 1 2 1 2 11 2 11 2 11 2 11 1 4/ 2 4/ 2 1 1 0 >= ⋅ − =⋅= ⋅ + =⋅= +=⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ −= −=⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ += = − − − kfora jea jea j j a j j a a k j j π π 051:080 Bioelectrical Design Revised: 01/29/2005 O. Poroy APERIODIC, CONTINUOUS-TIME SIGNALS AND THE FOURIER TRANSFORM Topics: The Fourier Transform Pair, properties of the Fourier Transform Fourier Analysis is not limited to periodic functions. LTI systems with aperiodic, continuous-time signals can be analyzed by a pair of equations known as the Fourier Transform Pair: ∫ ∫ ∞+ ∞− − +∞ ∞− ⋅= ⋅⋅= dtetxX deXtx tj tj ω ω ω ωω π )()( )( 2 1)( The Fourier Transform Pair Note the similarity between the Fourier Transform Pair and the Analysis and Synthesis Equations for periodic functions. The fixed frequency of ω0 is replaced by the variable frequency ω, because instead of a series of harmonic complex exponentials (with frequencies kω0) a function X(ω) is used to express x(t). Because ω is continuous, the summation in the Synthesis Equation is replaced by an integration. X(ω) is known as the Fourier Transform of x(t) and x(t) is called the Inverse Fourier Transform of X(ω). Example 6.2: Find the Fourier Transform of )()( tuetx at ⋅= − , where u(t) is the unit step function. ( ) ( ) ∞= = +− +∞ +− +∞ ∞− −− +∞ ∞− − ⋅ + − = =⋅⋅=⋅= ∫∫∫ t t tja tjatjattj e ja X dtedtetuedtetxX 0 0 1)( )()()( ω ωωω ω ω ω If a > 0: ω ω ω ω ja X ja X + = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ + −−= 1)( 10)( 051:080 Bioelectrical Design Revised: 01/29/2005 O. Poroy Properties of the Fourier Transform: If the Fourier Transform of a function x(t) or the inverse Fourier Transform of a function X(ω) exists, it can be calculated using the Fourier Transform Pair, but this may not always be necessary. There exist extensive tables containing the Fourier Transform of large numbers of functions and most books on Systems Theory or Signal Processing contain a short table with the most common functions. Often, the function to be transformed can be found in one of these tables. In other cases, the Fourier Transform of a function can be deduced from the Fourier Transform of a related function found in a table, by applying the properties of the Fourier Transform described in this section. Linearity: )()()()( )()( )()( 2121 22 11 ωω ω ω XbXatxbtxathen Xtxand XtxIf F F F ⋅+⋅⎯→←⋅+⋅ ⎯→← ⎯→← Time and Frequency Shift: )()( )()( )()( 0 0 0 0 ωω ω ω ω ω −⎯→←⋅ ⎯→←− ⎯→← − Xtxeand Xettxthen XtxIf Ftj tjF F Differentiation: ω ω ωω ω d Xdtxtjand Xj dt txdthen XtxIf F F F )()( )()( )()( ⎯→←⋅− ⋅⎯→← ⎯→← Integration: ∫ ∫ ∞− ∞− ⎯→←⋅⋅+⋅ − ⋅⋅+⋅⎯→← ⎯→← ω ξξδπ ωδπω ω ττ ω dXtxtx tj and XX j dxthen XtxIf F F t F )()()0()(1 )()0()(1)( )()(