Fourier Concepts ES3

© 2001 KEDMI Scientific Computing. All Rights Reserved. Square wave example: V(t)= 4/  sin(t) + 4/3  sin(3t) + 4/5  sin(5t) + 4/7  sin(7t) + 4/9  sin(9t) + … time Any periodic waveform can be constructed by adding sine and cosine waves with frequencies that are integer multiples of the waveform’s frequency. added frequency total signal … frequency |v| fofo 5f o 7f o 9f o 11f o 3f o Frequency spectrum of a square wave

Flute

Piano 4, 5, 6 f o = 260 Hz 2f o = 520 Hz 3f o = 780 Hz

Acoustic Bass fofo 2f o fofo

Hearing and the Ear a simplified explanation “unwound” cochlea (3.5cm) ear drum 030,000 Hair cell number amplitude of vibration = sin(2  200t) + 0.2sin(2  5000t) 200Hz 5kHz 1961 Nobel Prize, Georg von Bekesy

Hearing and the Ear (cont.)

FFT demonstration A microphone creates a voltage that is proportional to the pressure of a sound wave. This voltage is converted to a sequence of numbers that are stored in the computer’s memory The Fast Fourier Transform (FFT) is used to extract the amplitude of each sinusoid from the sound signal.

Filters

Filters are usually represented as two-port networks Two-port Network V in (t) + V out (t) Input PortOutput Port

Filters are usually represented as two-port networks Transfer Function: H(  ) V IN + V OUT Input PortOutput Port

Transfer Function Example INPUT Signal, v in (t)

FFT of the INPUT Signal: V IN (  ) Transfer Function Example

H(  )  1 Transfer Function Example The Transfer Function, H(  ) …a low-pass filter

Transfer Function Example H(  )  1 At 530 Hz: V OUT = V IN x 1 At 1060 Hz: V OUT = V IN x 1 At 1590 Hz: V OUT = V IN x 0.25 At 2120 Hz: V OUT = V IN x 0.0

Transfer Function Example FFT of the OUTPUT Signal: V OUT (  )

A real low-pass filter R 1 / j  C V in V out

The Transfer Function of a low-pass filter R=1000  C=1 uF  =1/RC is the cut-off frequency |H(  )|=1/√ rad/s = 159 Hz

The Transfer Function of a low-pass filter 45 o phase  = 1/RC