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LECTURE 16: FOURIER ANALYSIS OF CT SYSTEMS

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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:

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?

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!

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.

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, ’-’);

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;

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:

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.

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.

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:

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

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);

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

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

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

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.