Complex

Sounds

Reading: Yost Ch. 4

Most sounds in our everyday lives are not

simple sinusoidal sounds, but are complex

sounds, consisting of a sum of many

sinusoids.

The amplitude and frequency content of

natural sounds varies with time.

A spectrogram is a visual representation

of the spectrum of frequencies in a sound

(or other signal) as they vary with time.

Natural Sounds

There are two ways to describe a sound

wave: in the time domain and in the

frequency domain.

• Time domain: describes how the

instantaneous sound pressure/intensity

varies as a function of time.

• Frequency domain: describes sounds in

terms of the individual sinusoids that are

added together to produce the sound, in

the form of amplitude- and phase

spectra as a function of frequency.

Time vs. Frequency Domain

Fourier analysis can be used to convert

between the time and frequency domain

representations of a sound. In particular,

Fourier’s theorem states that all time domain

sounds are composed of a sum of sinusoids

of different frequencies, amplitudes and

phases. • If the signal is periodic, then the frequencies of

the sinusoids are harmonically related (integer

multiples of the fundamental frequency, f0=1/Pr)

• If the signal is aperiodic, then the frequencies

are continuous

Fourier Analysis 1

Pr

f0 = 1/Pr

. . .

frequency (Hz)

line spectrum

continuous

spectrum

If the amplitude variation of a complex sound is

important, then the waveform is described in

the time domain. If the frequency content is

important, then the amplitude and phase

spectra are described in the frequency domain.

One representation is obtained from the other

(time-frequency) using the Fourier transform

or inverse Fourier transform , respectively.

Fourier Analysis 2

The first term in the Fourier

series for the square wave is

shown in purple.

The second term in the Fourier

series for the square wave is

shown in purple; the sum is red.

The third term in the Fourier

series for the square wave is

shown in purple; the sum is red.

The sum of the first 10 terms is

shown in red. It looks very like a

square wave with some bumpiness.

• Tone burst: “gated” sinusoid with discrete

onsets and offsets.

• Spectrum of TB is continuous: extends over

a wider frequency range than infinitely long

tone (which has a line spectrum).

• Onsets and offsets add components, cause

“spectral splatter”

• Splatter around tone frequency heard as

onset and offset clicks when tone burst is

brief.

• Tone burst more clearly tonal and less click-

like the longer it is on, because the

proportion of Etotal at the frequency of

dominant sinusoid increases with duration.

• Splatter reduced by “shaping” rise/fall time

with gradual onsets and offsets to make it

inaudible.

Tone Bursts

Amplitude and Frequency Modulation

Two direct ways to change a simple

stimulus into a complex one are to

alter the amplitude and frequency of

the sinusoid with time.

Modulation means varying some

aspect of a continuous wave carrier

with an information-bearing

modulation waveform.

• In amplitude modulation (AM), the

amplitude or "strength" of the

carrier oscillations is varied with

time.

• In frequency modulation (FM), the

instantaneous frequency of the

carrier is varied with time.

Amplitude of a signal changes in a

sinusoidal manner over time (SAM).

Start with a sine-wave “carrier” (where

fc = carrier frequency)

D(t) = A sin(2πfct)

Let carrier amplitude (A) vary in a

sinusoidal manner over time:

A(t) = [1 + m sin(2πfmt)]

where fm = modulation frequency, and

m = modulation amplitude (0 ≤ m ≤ 1).

Therefore: D(t) = (A •A(t)) sin(2πfct)

D(t) = A [1 + m sin(2πfmt)] sin(2πfct)

Sinusoidal Amplitude Modulation 1

10

0

2

0

1 m

Top: SAM tone, showing “fine structure”

(waveform) of carrier (fc) varying in

amplitude at a modulation rate of fm.

Bottom: Amplitude (left) and phase (right)

spectra of SAM tone above.

Amplitude spectrum: fc is flanked by

two sideband frequencies at

(fc + fm) and (fc – fm).

Sideband amplitudes are equal to

A(m/2), where m is the amplitude of

modulator.

Sinusoidal Amplitude Modulation 2

0

90

-90

tone SAM

tone

Frequency (f) of a signal varies with time; often

in a linear (LFM) or sinusoidal manner (SFM).

LFM: f varies in a linear manner with time from

a starting frequency to an ending frequency in

an interval t:

D(t) = A sin(2πft)

f = fstart + kt, where k = (fend – fstart) / T

The parameter k is call the chirp rate.

SFM: f varies in a sinusoidal manner in time

f = fc + m sin(2πfmt), where m = fmax –fmin

M is the modulation depth and fm is the

modulation rate (δf/t in Hz/s)

The spectra of FM stimuli changes over time;

changes in frequency (and amplitude) over time

can be graphed using spectrograms.

SFM

LFM

“Spectrogram” AM/FM

Frequency Modulation

SFM

The addition of two sinusoids of different

frequencies also produces a complex

sound. If the two tones have very different

frequencies, then they add like harmonics

in a series (where the frequencies can be

seen as the time separations (1/f1 and

1/f2) between peaks).

Beats

If the tones are close in frequency, then

the waveform appears to be a single tone

with a sinusoidal amplitude modulation

(similar to, but not the same, as SAM)

• Frequency = mean = (f1+f2)/2

• Amplitude of waveform beats at a

rate equal to f1-f2

1/f1

1/f2

1/(f1-f2) 1/[(f1+f2)]/2

A square wave is equivalent to turning a

sound on and off in a periodic manner

such that the on time is the same as the

off time. Pulse on time = pulse off time

(50% “duty cycle”).

Square waves have spectral energy at a

fundamental frequency f0 = 1/Pr, and at

odd harmonics of f0: 3f0, 5f0, 7f0 etc..

Amplitude of each higher harmonic =

1/harmonic number:

3rd harmonic (3f0) = 1/3(Af0)

5th harmonic (5f0) = 1/5(Af0), etc.

Pr = 2 ms

F0 = 500 Hz

1500 Hz 2500 Hz

Square Waves

In studying hearing, it is often useful to

present a very brief acoustic click. Transients

(clicks) are very brief, non-sinusoidal

“impulses” that are the sum of many

sinusoids in phase at only one point in time.

Click spectra are distinctive:

• Amplitude spectrum: Continuous

with minima at frequencies equal

to integer multiples of 1/D, where

D = click duration in sec.

• Phase spectrum: 90° for all

frequencies.

D

1/D

2/D

3/D

Transients

f = 1/D

Click trains

• clicks repeating at a constant rate with

period Pr

Generate a line spectrum (like a harmonic

series) with discrete frequencies equal to

integer multiples of 1/Pr (i.e., the inter-click

interval).

Click duration (D) shapes the spectrum,

generating spectral notches at frequency of

1/D, 2/D, 3/D, etc.

“Repetition Pitch”: Click trains can give rise

to pitch perception because the temporal

structure and spectrum are similar to complex

tones.

Click Trains

scale

Repetition pitch is the sensation of tonality

in a sound, in which this tonal quality is

solely obtained due to a repeated pattern,

rather than a sinusoidal waveform.

The effect is perceived most prominently if

the repeated sound contains a wide

spectrum of frequencies, like clicks.

“Voiced” sounds (e.g., vowels) are

basically click trains produced by larynx,

with spectra shaped by vocal tract.

Amplitude spectra of three

vowels in human speech

Repetition Pitch

By definition, noise is a sound whose

instantaneous amplitude (A) varies

randomly in time.

Gaussian noise: A varies probabilistically

according to normal (“Gaussian”)

distribution.

Phase also varies randomly in Gaussian

noise.

Noise

• Total noise power (TP): Sum of

amplitudes of all sinusoids (i.e.,

bandwidth x intensity).

• “Spectrum level” (N0):

Average power per unit bandwidth

(i.e., average intensity in a band of

noise 1 Hz wide):

– N0 = TP/BW

= TP (dB) – 10 log BW

– For the figure:

N0 (dB) = 50 dB

= 80 dB – 10 log(1000)

Noise Intensity

• White noise:

– power spectrum is flat across all

frequencies.

– Bandwidth is very broad.

• Pink noise:

– Average power level drops 50%

(3 dB) per octave.

– Therefore, power level is constant

ratio of center frequency to BW.

“Colors” of Noise

Noises are usually broadband, containing

a large range of frequencies, but can also

be narrowband containing a limited

number of frequency components.

Narrowband noises have an envelope that

fluctuates in proportion to noise bandwidth.

That is, the rate of amplitude modulation

increases with increasing bandwidth:

• Left (a): BW = 10 Hz

• Right (b): BW = 25 Hz

Narrowband Noise

Narrow-band

(band limited) noise

Many signals can be characterized by

an envelope and the fine-structure

waveform that falls under the envelope

(e.g., SAM tones). In fact, most

waveforms can be described by the

following formula:

x(t) = e(t)f(t)

where e(t) is the envelope function, f(t)

is the fine-structure waveform, and x(t)

is the complex waveform.

SAM noise:

x(t) = [1 + m sin(2πFmt)] n(t)

Envelopes