BME 315 Exam 2 Concepts

How do we find fourier coefficients, a_k, for any periodic signal?

Fourier analysis integral

Go from x(t) to a_k

Defined over 1 period, T₀

What are full-wave rectified sine waves (FWRS)?

Found in DC power supplies

Found in AC to DC power converters

Recharges batteries

What is the Fourier Analysis of a FWRS?

1. Use fourier integral for FWRS to find a_k

2. Plug in FWRS signal

3. Plug in T₀

4. Euler’s formula to write as sum of two integrals

5. Perform integral

6. Plug in upper and lower bounds

7. find e value

How do you determine the DC value of a Fourier Series

1. Set k = 0

2. Solve for a_k

What is the spectrum of a FWRS?

Use a_k values for the spectrum

Lines go at multiples for F₀

Length of lines is a_k

Spectrum goes both directions

How do finite and infinite synthesis compare?

Finite is bounded while infinite is not

Infinite has an infinite number of spectral lines and infinite bandwidth

Magnitude of the very high harmonics are extremely small

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What is the point of the Fourier Series Integral?

We can represent any periodic signal by using a sum of complex exponential signals

How can you derive Fourier Series Coefficients for pulse waves?

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Duty cycle

r = τ /T₀

Ratio of time a signal or system spends active (on) compared to total duration of complete repeating period

How does the fourier series coefficients change with a square wave?

τ = T₀/2 : Wave is square

a_k = 0 when k is odd and phase is 0 because x(t) is centered at t=0

Fourier series coefficients are discrete frequencies kF₀

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How can you derive Fourier Series Coefficients for triangular waves?

Continuous signals over one period

User fourier series integral to find a_k

x(t) = |2t/T_o| for -T₀/2 < t ≤ T₀/2

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How can you derive Fourier Series Coefficients for half-wave recified sinusoids?

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How can you draw the spectrum from the Fourier Series Coefficients?

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How do we compute signal power?

A spectrum plot of a_k is the distribution of signal power vs frequency

average power is the total energy of one period divided by the period

power depends only on amplitude of signal

typically measured in dB

How do we convert continuous-time signals to digital signals that can be processed by a computer?

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How do we compute the values of a discrete time signal directly from a formula?

we can use the formula

can also be obtained by sampling

How do we plot sequences?

we use a stem format

How do we sample sinusoidal signals?

discrete-time cosine signal

normalized radian frequency or digital frequency

What is aliasing?

alias: two different discrete-time sinusoid formulas can define the same signal values

What is the sampling theorem?

a continuous-time signal with frequencies no higher than f_max can be constructed from its samples if the samples are taken at a rate f_s = 1/T_s, that is greater than 2f_max (Nyquist rate)

typically use the lowest possible sampling rate to minimize cost of storage and processing

if there are not enough samples, aliasing occurs and reconstruction becomes difficult

What happens when we over-sample?

when we sample at a rate of over 2f_max

use principal alias to reconstruct signal

What happens when we under-sample?

when f_s < 2f₀

aliasing occurs

What are the frequencies of physiological signals?

Nyquist rate sampling f_s > 2f_max

What is the purpose of a D to C converter?

interpolates a smooth continuous-time function y(t) through the given discrete-time samples y[n]

How does a D to C converter work?

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What is a zero-order hold interpolation?

simplest pulse shape is a symmetric square pulse

What is first-order interpolation?

triangular pulse consists of first-order polynomial segments (straight lines)

What is cubic spline interpolation?

third-order polynomial segments

How does oversampling aid interpolation?

we get better reconstructions

What is band limited interpolation?

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What is a digital signal processor?

DSP

measure, filter, compress continuous real-world analog signals

What is a Finite Input Response (FIR) filter?

DSP whose impulse response settles to zero in a finite amount of time

What is a discrete-time system?

a sequence of data values defined only at specific points rather than continuously

What is the running average filter?

running or moving average of two or more consecutive values of the sequence

What are our options for indexing the output signal?

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What is the general FIR filter?

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What are the important properties of FIR filters?

modify signals usefully

length of average intervals has a big effect

running average filters introduce a shift of M/2 samples

What is the unit impulse sequence?

What is the unit impulse response?

What are the three ways to represent the FIR filter?

use shifted impulses to write h[n]

plot the values h[n]

list values b_k

What is the convolution sum?

convolving the sequence h[n] and x[n] for finite length

h[n] = 0 for n<0 and n>M

What is the length of a convolution?

How do we use a FIR filter and convolution in MATLAB?

How do we filter the Unit-Step signal?

unit step turns on at n=0

3 cases for the output

What are the properties of convolution?

convolution is communicative

convolution is associative and distributive

What do we need to compute a FIR filter?

on a computer chip

multiplier

adder

unit delay operator

How do we represent systems with block diagrams?

nodes: junction points

branches: connections between nodes

feed forward difference equation

direct form: standard structure

How do we test for time invariance?

look at examples

How do we test for Linearity?

look at examples

What are cascaded LTI systems?

cascade connection of two systems means the output of the first is the input to the second system

for LTI, order of systems can be rearranged and provide the same output