## Musimathics: The Mathematical Foundations of Music, Volume 2 |

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Page 210

We see that H describes a complex vector path from 2 to 0 as frequency goes

from 0 Hz to the Nyquist frequency. The function H(eiwT) given in equation (5.20)

is called the

transferred to y depending on frequency. Note that the function does not depend

upon the passage of time; the parameter n does not appear in it anywhere.

Hence, the

upon 0), the ...

We see that H describes a complex vector path from 2 to 0 as frequency goes

from 0 Hz to the Nyquist frequency. The function H(eiwT) given in equation (5.20)

is called the

**transfer function**of the filter because it shows how much of x istransferred to y depending on frequency. Note that the function does not depend

upon the passage of time; the parameter n does not appear in it anywhere.

Hence, the

**transfer function**is time-invariant. The function H(e"0T) only dependsupon 0), the ...

Page 211

There are methods to circumvent the need for four-dimensional graphics by

showing how the function transforms sets of lines that lie in the complex plane.

Each line is mapped into a corresponding curve in the complex plane, and these

can be represented in two dimensions. This is the approach taken in figure 5. 10,

but this is a simple filter, and this approach will not be powerful enough for more

complicated functions. It would be nice to reduce the complexity of the

There are methods to circumvent the need for four-dimensional graphics by

showing how the function transforms sets of lines that lie in the complex plane.

Each line is mapped into a corresponding curve in the complex plane, and these

can be represented in two dimensions. This is the approach taken in figure 5. 10,

but this is a simple filter, and this approach will not be powerful enough for more

complicated functions. It would be nice to reduce the complexity of the

**transfer****function**to ...Page 212

Magnitude of the

real frequency variable ft) Remembering that taking the angle of a complex

number reveals its phase, let's define the phase response for frequency a) as the

angle of the

5.23) 0(G)) is a real-valued function of the real variable frequency ft). 5.4.8

combination of ...

Magnitude of the

**Transfer Function**(5.22) G (ft)) is a real-valued function of thereal frequency variable ft) Remembering that taking the angle of a complex

number reveals its phase, let's define the phase response for frequency a) as the

angle of the

**transfer function**: Q(0)) = ZH(e"°T). Angle of the**Transfer Function**(5.23) 0(G)) is a real-valued function of the real variable frequency ft). 5.4.8

**Transfer Function**Finally, we must show that the**transfer function**consists of thecombination of ...

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### Contents

Digital Signals and Sampling | 1 |

Musical Signals | 49 |

Spectral Analysis and Synthesis | 103 |

Copyright | |

10 other sections not shown

### Other editions - View all

Musimathics: The Mathematical Foundations of Music, Volume 2 Gareth Loy,John Chowning Limited preview - 2011 |

### Common terms and phrases

acceleration acoustical aliasing allpass filter amplitude angle audio band bandwidth binary coefficients complex number complex plane components constant convolution convolved corresponding cosine wave counterclockwise defined delay line derivative discrete displacement dissipation encoding energy example force Fourier transform frequency domain frequency response fundamental analysis harmonic Hilbert transform imaginary number impedance impulse response impulse train increases infinite input signal integer inverse length linear lowpass filter magnitude modulation motion multiply negative frequencies noise nonlinear Nyquist frequency oscillator output period phasor positive frequencies pressure probe phasor quantization quency radians range ratio reactance real number rectangular function reflected resonant result rotation sampling rate scaled sequence shown in figure shows sidebands sine wave sinusoid sound spectral spectrum STFT string synthesis theorem timbre transfer function tube unit circle vector velocity vibration volume wave equation waveform waveguide Xk(n Z transform zero