FFT and Related Issues

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FFT and Related Issues

• By Marco Congedo

In Medio stat Virtus
Qwesdfgj
AGA!
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Some basic concepts in DSP

• Shannons Sampling Theorem
• Nyquist frequency (Maximal frequency contained in
  an analog signal)
• Nyquist Rate (Minimum SR required)
• Folding frequency (Maximal frequency that can be
  represented, ½ Sample Rate)
• Aliasing
• Quantization Noise (depends on the bits)

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Effect of Aliasing
Sampling the analog signal causes the original
spectrum to repeat around multiples of the
sampling frequency
a Spectrum of an Analog Speech Signal b After
Digital Sampling. Sample rate 8000 Hz c After
Digital Sampling. Sample rate 6000 Hz d After
Digital Sampling. Sample rate 5000 Hz
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Quantization Noise

• The Quantization Noise refers to the noise added
  to the digital signal because of the
  representation of a continuous Analogue signal by
  means of a discrete number. The Analogue-Digital
  Converter (ADC) in fact can represent each
  analogue value only by one of the 2b discrete
  value allowed by the number of bits used (b). The
  quantization noise decrease as the number of bits
  increase.

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EEG Analogue and Digital DA
The Digital Sampling Method has the advantage
that the Amplitude values are recorded in time
in a computer. These value are thus ready to be
analyzed offline
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Sine and Cosine Components
Any Sinusoidal Waveform can be considered as the
sum of two waves of the same frequency separated
in time by a quarter of a cycle. These wavea are
called the Sine and Cosine Components
(c) is equal to a cosine wave (a) plus a sine
wave (b)
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Product of Signals and Sine Waves
When two sine-like waves are multiplied together
(one or more cycle must be involved in the
product), the mean cross-product SXY/n is finite
(ltgt0) when their frequencies are the same and
zero when their frequencies are different. Notice
that SXY/n is nothing more than the Covariance
divided by n
a
c
b
d
a Product of a sine at 9.2 Hz with itself b
Product of a sine at 9.2 Hz and a sine at 6 Hz c
Product of an EEG epoch (2.5 sec) and a sine at
9.2 Hz d Product of an EEG epoch (2.5 sec) and a
sine at 6 Hz
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DFT a Formalization
DFT (Time to Freq. Domain)
I DFT (Freq. To Time Domain)
Real (Cosine) Component
Imaginary (Sine) Component
MAGNITUDE
PHASE
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A Geometrical View
Cosine
Power
Sine
F
B
Magnitude
Sine
G
Sin2 Cos2 Magnitude
Cosine
C
A
ABSine ACCosine BCMagnitude Mag(AB2 AC2)½
BCGFPower PowMag2 Angle ACBPhase Phase
Arctg(AB/AC)
Sin2 Cos2 Power
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Spectral Leakage and Windows
The Rectangular Window represents accurately
those frequecies components that corrispond
exactly to the discrete Friquencies of the DFT
(1). For these frequencies the Hamming Window
produce some leakage (2). For frequencies in
between two discrete freqencies of the DFT the
rectangular window produce severe distorsions (3)
that are still negligeable when a Hamming Window
is applaied (4). Tapered Windows reduce Spectral
Leakage at the expense of some broadening around
individual spectral components.
1
2
Sine 12 Hz Rectangular Window
Sine 12 Hz Hamming Window
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3
Sine 12.5 Hz Rectangular Window
Sine 12.5 Hz Hamming Window
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Effect of Windows on a complex Signal
The use of Tapering Windows aims to minimize both
Spectral Leakage and Broadening
Rectangular (doing-nothing)
Triangular
Hamming
Hamming tapering only the ends of data
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FFT and Related Issues

• .End

In Medio stat Virtus
Qwesdfgj
AGA!