Home > A Short Introduction to Signal Processing

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Tuomas Puoliväli tuomas.a.b.puolivali@student.jyu.fi 20th of February 2013

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► A signal is a function of an independent variable such as time,

distance, position, or temperature. Some examples of biomedical signals are:

▶ Electrocardiogram (ECG), electroencephalogram (EEG) and

magnetoencephalogram (MEG)

► A signal is said to be continuous when its domain is the set of

real numbers, and discrete otherwise

► Discrete signals are presented as sequences of numbers called

samples

► An analog signal is a real-valued continuous signal ► A digital signal is discrete in time and value

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0 2 4 6 8 10 12 14 −1 −0.5 0 0.5 1

Time / s Amplitude

Sine Wave

Continuous Signal 0 2 4 6 8 10 12 14 −1 −0.5 0 0.5 1 Discrete Signal 0 2 4 6 8 10 12 14 −1 −0.5 0 0.5 1 Digital Signal

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► Signal processing usually refers to:

▶ Signal generation ▶ Modifying signals ▶ Extracting information from signals

► Signal processing benefits from improvements in the areas of

electrical engineering, applied mathematics, statistics, mathematical information technology, ..

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► ��Frequency is the number of occurrences of a repeating event

per unit time�� –Wikipedia

► The SI unit of frequency is hertz (symbol Hz, 1 Hz = 1 / s)

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► Sampling is the process of converting a continuous signal to a

discrete one

► Sampling rate, usually denoted by fs, is the number of samples

per second collected from a continuous signal

▶ Sampling rate is given in the unit of hertz

► ��If a function x(t) contains no frequencies higher than B hertz, it

is completely determined by giving its ordinates at a series of points spaced 1

2B

seconds apart�� –Shannon

▶ For example, consider the human hearing sense. The human

hearing range is about from 20 Hz to 20 kHz, so the sampling frequency of audio signals must be at least 40 kHz to include all audible frequencies (audio compact discs use 44.1 kHz, telephones 16 kHz)

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► Signals are usually studied in

▶ time-domain (with respect to time) ▶ frequency-domain (with respect to frequency) ▶ time and frequency domains simultaneously, using some

time-frequency representation (TFR)

► Fourier transforms can be used to transform signals from

time-domain to frequency-domain, and vice versa

► Time-frequency representations can be computed using

short-time Fourier transform (STFT) or wavelets

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► The frequency-domain representation of a digital time-domain

signal x[t] can be calculated using the discrete Fourier transform (referred as DFT or the analysis equation), defined as: X[k] =

N−1

��

n=0

x[n]e−2��ik n

N

(1) =

N−1

��

n=0

x[n][cos(−2��k n N ) + isin(−2��k n N )] (2) where 0 �� k �� N − 1 and i is the imaginary unit. The equation (2) is obtained through the Euler��s formula ei�� = cos(��) + isin(��) (3)

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► The time-domain representation of a frequency-domain signal

X[k] can be calculated using the inverse discrete Fourier transform (referred as IDFT or the synthesis equation), defined as: x[n] = 1 N

N−1

��

k=0

X[k]e−2��ik n

N

(4) (5) where 0 �� n �� N − 1 and i is the imaginary unit.

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► Fast Fourier transform (FFT) is ��the most important algorithm of

our lifetime��. It is needed at least for:

▶ Signal processing (convolution, digital filters) ▶ Fast multiplication of large integers ▶ Solving partial differential equations ▶ Magnetic resonance imaging (MRI)

► The computational complexity of the discrete Fourier transform is

O(N2) where N is the signal length

▶ FFT produces the exact same result in O(N log N) operations

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x0 x4 x2 x6 x1 x5 x3 x7 X0 X1 X2 X3 X4 X5 X6 X7 �� �� �� �� �� �� �� ��

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► DFT is complex valued �� carries information about magnitude

and phase

► The magnitude spectrum of a frequency-domain signal X is

given by its absolute value |X|

► The phase spectrum of a frequency domain-signal X is given by

it argument arg(X)

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0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 −4 −2 0 2

Time / s Amplitude / mV

Normal ECG

0 10 20 30 40 50 60 70 0 50 100

Frequency / Hz

Magnitude

Magnitude Spectrum

0 10 20 30 40 50 60 70 −200 −100 0 100

Phase / Rad Frequency / Hz

Phase Spectrum

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► Discrete convolution of signals f and g is defined as:

(f ∗ g)[n] =

��

��

m=−��

f[n] · g[n − m] (6) =

��

��

m=−��

f[n − m] · g[m] (7)

► The resulting new signal is usually viewed as a modified or

filtered version of one of the original signals

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► Convolution has an important property known as convolution

theorem. Given two signals f and g the convolution theorem states that F{f ∗ g} = F{f} · F{g} (8) where F is used to denote DFT. By applying IDFT, denoted by F

−1, on the both hand sides of the equation (6) one gets

f ∗ g = F−1{F{f} · F{g}} (9) so it is possible to calculate convolution efficiently by using Fourier tranforms.

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► Low-pass filters

▶ Pass low frequencies, attenuate high frequencies

► High-pass filters

▶ Pass high frequencies, attenuate low frequencies

► Band-pass filters

▶ Pass frequencies within a specified range, attenuate frequencies

outside that range

► Band-stop filters

▶ Attenuate frequencies within a specified range, pass frequencies

outside that range

► If a filter is linear and time-invariant, it is completely

characterized and best described by its frequency response

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► If a filter is linear and time-invariant, its frequency response is

given by either

▶ F{y}/F{x}, where x is the input signal and y the output signal ▶ By taking the Fourier transform of the impulse response of the filter,

which is the filter��s response to Kronecker��s delta ��(n) = { 1, n = 0 0, n = 0

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