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LECTURE 16: FOURIER ANALYSIS OF CT SYSTEMS. Objectives: Response to a Sinusoidal Input Frequency Analysis of an RC Circuit Response to Periodic Inputs Response to Nonperiodic Inputs Analysis of Ideal Filters Resources: Wiki: The RC Circuit CN: Response of an RC Circuit CNX: Ideal Filters.

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LECTURE 16: FOURIER ANALYSIS OF CT SYSTEMS

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Lecture 16 fourier analysis of ct systems

LECTURE 16: FOURIER ANALYSIS OF CT SYSTEMS

  • Objectives:Response to a Sinusoidal InputFrequency Analysis of an RC CircuitResponse to Periodic InputsResponse to Nonperiodic InputsAnalysis of Ideal Filters

  • Resources:Wiki: The RC CircuitCN: Response of an RC CircuitCNX: Ideal Filters

URL:


Lecture 16 fourier analysis of ct systems

Differential Equations

  • For CT systems, such as circuits, our principal tool is the differential equation.

  • For the circuit shown, we can easily compute the input/output differential equation using Kirchoff’s Law.

  • What is the nature of the impulse response for this circuit?


Lecture 16 fourier analysis of ct systems

Numerical Solutions to Differential Equations

  • Consider our 1st-order diff. eq.:

  • We can solve this numerically by setting t = nT:

  • The derivative can be approximated:

  • Substituting into our diff. eq.:

  • Let and :

  • We can replace n by n-1 to obtain:

  • This is called the Euler approximation to the differential equation.

  • With and initial condition, , the solution is:

  • The CT solution is:

  • Later, we will see that using the Laplace transform, we can obtain:

  • But we can approximate this:

  • Which tells us our 1st-order approximation is accurate!


Lecture 16 fourier analysis of ct systems

Higher-Order Derivatives

  • We can use the same approach for the second-order derivative:

  • Higher-order derivatives can be similarly approximated.

  • Arbitrary differential equations can be converted to difference equations using this technique.

  • There are many ways to approximate derivatives and to numerically solve differential equations. MATLAB supports both symbolic and numerical solutions.

  • Derivatives are quite tricky to compute for discrete-time signals. However, in addition to the differences method shown above, there are powerful methods for approximating them using statistical regression.

  • Later in the course we will consider the implications of differentiation in the frequency domain.


Lecture 16 fourier analysis of ct systems

Series RC Circuit Example

Difference Equation:

R=1;C=1;T=0.2;

a=-(1-T/R/C);b=[0 T/R/C];

y0=0; x0=1;

n=1:40;

x=ones(1,length(n));

y1=recur(a, b, n, x, x0, y0);

Analytic Solution:

t=0:0.04:8;

y2=1-exp(-t);

y1=[y0 y1];

n=0:40;

plot(n*T, y1, ’o’, t, y2, ’-’);


Lecture 16 fourier analysis of ct systems

Example: RC Circuit

  • Using our FT properties:

  • Compute the frequency response:

  • RC = 0.001;

  • W=0:50:5000;

  • H=(1/RC)./(j*w+1/RC);

  • magH=abs(H);

  • angH=180*angle(H)/pi;


Lecture 16 fourier analysis of ct systems

Response of an LTI System to a Sinusoid

  • Consider an LTI CT system with impulse response h(t):

  • We will assume that the Fourier transform of h(t) exists:

  • The output can be computed using our Fourier transform properties:

  • Suppose the input is a sinusoid:

  • Using properties of the Fourier transform, we can compute the output:


Lecture 16 fourier analysis of ct systems

Example: RC Circuit (Cont.)

  • We can compute the output for RC=0.001 and ω0=1000 rad/sec:

  • We can compute the output for RC=0.001 and ω0=3000 rad/sec:

  • Hence the circuit acts as a lowpass filter. Note the phase is not linear.

  • If the input was the sum of two sinewaves:

  • describe the output.


Lecture 16 fourier analysis of ct systems

Response To Periodic Inputs

  • We can extend our example to all periodic signals using the Fourier series:

  • The output of an LTI system is:

  • We can write the Fourier series for the output as:

  • It is important to observe that since the spectrum of a periodic signal is a line spectrum, the output spectrum is simply a weighted version of the input, where the weights are found by sampling of the frequency response of the LTI system at multiples of the fundamental frequency, 0.


Lecture 16 fourier analysis of ct systems

Example: Rectangular Pulse Train and an RC Circuit

  • Recall the Fourier series fora periodic rectangular pulse:

  • Also recall the system response was:

  • The output can be easily written as:


Lecture 16 fourier analysis of ct systems

Example: Rectangular Pulse Train (Cont.)

  • We can write a similar expression for the output:

1/RC = 1

  • We can observe the implications of lowpass filtering this signal.

  • What aspects of the input signal give rise to high frequency components?

  • What are the implications of increasing 1/RC in the circuit?

  • Why are the pulses increasingly rounded for lower values of 1/RC?

  • What causes the oscillations in the signal as 1/RC is increased?

1/RC = 10

1/RC = 100


Lecture 16 fourier analysis of ct systems

Response to Nonperiodic Inputs

  • We can recover the output in the time domain using the inverse transform:

  • These integrals are often hard to compute, so we try to circumvent them using transform tables and combinations of transform properties.

  • Consider the response of our RC circuit to a single pulse:

  • MATLAB code for the frequency response:

  • RC=1;

  • w=-40:.3:40;

  • X=2*sin(w/2)./w;

  • H=(1/RC)./(j*w+1/RC);

  • Y=X.*H;

  • magY=abs(Y);


Lecture 16 fourier analysis of ct systems

Response to Nonperiodic Inputs (Cont.)

  • We can recover the output using the inverse Fourier transform:

  • syms X H Y y w

  • X = 2*sin(w/2)./w;

  • H=(1/RC)./(j*w+1/RC);

  • Y=X.*H;

  • Y=ifourier(Y);

  • ezplot(y,[-1 5]);

  • axis([-1 5 0 1.5])

1/RC = 1

1/RC = 1

1/RC = 10

1/RC = 10


Lecture 16 fourier analysis of ct systems

Ideal Filters

  • The process of rejecting particular frequencies or a range of frequencies is called filtering. A system that has this characteristic is called a filter.

  • An ideal filter is a filter whose frequency response goes exactly to zero for some frequencies and whose magnitude response is exactly one for other ranges of frequencies.

  • To avoid phase distortion in the filtering process, an ideal filter should have a linear phase characteristic. Why?

  • We will see this “ideal” response has some important implications for the impulse response of the filter.

  • Highpass

  • Lowpass

  • Bandstop

  • Bandpass


Lecture 16 fourier analysis of ct systems

Ideal Linear Phase Lowpass Filter

  • Consider the ideal lowpass filterwith frequency response:

  • Using the Fourier transform pairfor a rectangular pulse, and applyingthe time-shift property:

  • Is this filter causal?

  • The frequency response of an idealbandpass filter can be similarly defined:

  • Will this filter be physically realizable?Why?

  • PhaseResponse

  • ImpulseResponse


Lecture 16 fourier analysis of ct systems

Summary

  • Showed that the response of a linear LTI system to a sinusoid is a sinusoid at the same frequency with a different amplitude and phase.

  • Demonstrated how to compute the change in amplitude and phase using the system’s Fourier transform.

  • Demonstrated this for a simple RC circuit.

  • Generalized this to periodic and nonperiodic signals.

  • Worked examples involving a periodic pulse train and a single pulse.

  • Introduced the concept of an ideal filter and discussed several types of ideal filters.

  • Noted that the ideal filter is a noncausal system and is not physically realizable. However, there are many ways to approximate ideal filters, and that is a topic known as filter design.


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