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i.e. audio

i.e. digital bitstream

Fundamental property of all analog signalsproperties

amplitude

Decompose

into summation

of sinusoids

phase

frequency

Fourier Transform

How do we analyze the frequency components of a complex signal

Time space x(t)

Frequency space X(w)

single frequency signal w0

w

t

w0

t

w

Some properties

- X(w) is complex -- complex conjugate encodes phase
- Fourier transform is invertable

Digital Signals

Sample amplitude at discrete time intervals

1,0

.55

.46

-,6

-1.0

Nyquist limit (http://www.medcyclopaedia.com)

(Harry Nyquist, 18891976, Swedish - American physicist), the maximum frequency of a signal that can be measured with a method that employs sampling of the signal with a specific frequency, the sampling frequency. According to Shannons sampling theorem, a signal must be sampled with a frequency at least twice the frequency of the signal itself. The maximum measurable frequency the Nyquist limit or frequency is thus half the sampling frequency. If the signal frequency is higher than the Nyquist limit, aliasing occurs.

Discrete Fourier Transform

Given a signal represented as a time sequence of samples, the DFT gives us a seqence of frequency/phase amplitudes

1,0

.55

.46

w

w0

-,6

-1.0

w

Signals and noise

- What is noise?
- Any signal other than the one you are interested in!
- Sources of Noise (the usual suspects)
- statistical signals from active electronic components
- crosstalk from other channels or other signals in the same channel
- signals sensed from external sources (power supply, EM radiation)

Trival Example

Signal to Noise ratio

The relative amplitude of the signal of interest o the noise signal

Signal of interest +

Noise signal =

Noisey Signal

Filtering out the Noise

Finite Impulse response (FIR) filter

Ideal Pulse (time domain)

Ideal Pulse (frequency domain)

Zero rise/fall time

to inifinity

to inifinity

w

t

Ideal Pulse (time domain)

actual rise/fall time

finite band of component frequencies

w

The number and values of the component freqencies is related to the rise/fall time of the pulse

http://www.chem.uoa.gr/Applets/AppletFourAnal/Appl_FourAnal2.html

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