8/11/2019 W4 L1 Slides

1/15

AC Signals - 2

Week 4 - Lecture 1 Mark Bocko

Topics: Tone and frequency spectra

Fourier Series and Fourier Spectral Analysis

Filtering and frequency content of signals

Guitar pickups

RLC band-pass circuits

1

8/11/2019 W4 L1 Slides

2/15

Tone?

The quality or character of sound

2

The characteristic quality or timbreof a

particular instrument or voice. The character or quality of a musical

sound or voice as distinct from its pitch

and intensity.Frequency !pitchSound pressure level !intensity (loudness)

Attack, Spectrum, Spectral evolution !tone (timbre)

8/11/2019 W4 L1 Slides

3/15

Building a Square Wave from sine waves

3

S(t) = sin[2!f0t] +

1

3sin[2!(3f

0)t] +

1

5sin[2!(5f

0)t] +...

Exploring tone by building different waveforms

Building a triangle wave from sine waves

T(t) = sin[2!f0t] +1

32 sin[2!(3f0 )t] +

1

52 sin[2!(5f0 )t] +...

Notice the 1/n2coefficients!

8/11/2019 W4 L1 Slides

4/15

Tone is related to the spectrum of a waveform.

4

Frequency

Amplitude

Fundamental(f0)

Overtones

The spectrum is a display of the frequency content of a waveform.

Formant (envelope)

If the overtones are at integer multiples of the fundamental theyare called harmonics, e.g., 2f0!2ndharmonic, etc.

Most musical sounds have harmonic (or nearly so) overtones.

8/11/2019 W4 L1 Slides

5/15

Fourier series

Build any periodic waveform (with period1/f0) from sine and cosine functions

5

x(t) =a

0

2

+ [ancos(2!nf

0t) +b

n

n=1

!

" sin(2!nf0t)]

Starting with a function y(t), you can computethe Fourier Coefficients (the recipe!) from

an =1

!y(t) cos(2!nf0t) dt n! 0

onecycle

"

bn =

1

!

y(t) sin(2!nf0t) dt n! 1

one

cycle

"

8/11/2019 W4 L1 Slides

6/15

Meaning of Fourier Coefficient integrals

6

time Amplitude

b1 (sin)

a1 (cos)

= 1

= 0

time

A

mplitude

1

-1

8/11/2019 W4 L1 Slides

7/15

7

Fourier Coefficient integrals are a measure of the alikeness of theoriginal function and each sine or cosine wave

time Amp

litude

b2 (sin)

a2 (cos)

= 0

= 0

time Amplitud

e

1

-1

8/11/2019 W4 L1 Slides

8/15

Spectral analysis

8

timeAmplitude

T

f0 = 1/T

2f0

3f0

4f0.

.

.

Lowest frequencyin spectrum

sines b1

b2

b3

b4

Theoretically this goes on to infinite frequency

8/11/2019 W4 L1 Slides

9/15

Spectral analysis continued

9

timeAmplitude

T

f0 = 1/T

2f0

3f0.

.

.

Lowest frequencyin spectrum

cosinesa1

a2

a3

Sn(f) = a

n

2+b

n

2( )1/2 Magnitude of the nth

spectral component

8/11/2019 W4 L1 Slides

10/15

Spectral Analysis Examples

Guitar

10

Contrabass saxophone

8/11/2019 W4 L1 Slides

11/15

Applying a filter to alter tone

11

Frequency

Amplitude

Back to circuits

Low pass

Examples

8/11/2019 W4 L1 Slides

12/15

8/11/2019 W4 L1 Slides

13/15

Electrical model of a guitar pickup

13

~

R

L

C

vs

RL

C 6000 turnsL "2 HenryC "120 pF

R "5,000 #(DC resistance)*

* Effective resistance at audio frequencies "10x DC resistance due to losses in the magnets

vs= voltage signalinduced in coil

from string motion

8/11/2019 W4 L1 Slides

14/15

Guitar pickup analysis: Find the output voltage, vout

14

Apply KVL:

!iR! j!Li + vs!

1

j!Ci = 0

i = vs

R + j!L +1

j!C

vout

= i 1

j!C=v

s

1

j!C

R + j!L +1

j!C

vout

=vs

1

1! !2 !r

2+ j! (!

rQ)

~R vsj!L

1

j!C

- +

vout

!r

2=

1

LC

Q =1

!rRC

=

1

2"

T

RC

Resonant frequency

Q quality factor

T is the period of theresonance of the circuit

vout

= vs

1

(1! !2 !r

2 )2 + !2 (!r

2Q

2 )"# $

%

1/2

i

8/11/2019 W4 L1 Slides

15/15

Frequency response of pickups

15

''=