Sound in Time and FrequencySound in Time and Frequency
Telecommunications 1Telecommunications 1P. MathysP. Mathys
Types of Sound SignalsTypes of Sound Signals
Sound Signals
Deterministic(predictable)
Random(unpredictable)
periodic aperiodic stationary non-stationary
sinesawtooth
rectangular
talking clock(date/time)
white noise speechmusic
sine_sig.wav wnoise_sig.wav speech_sig.wav201pm.wav
sine_sig.wav 201pm.wav speech_sig.wavwnoise_sig.wav
Characterization of TonesCharacterization of Tones
Sounds (digital and analog) are made Sounds (digital and analog) are made up from combinations of tones, e.g.,up from combinations of tones, e.g.,
Pure tones are also called sinusoids.Pure tones are also called sinusoids.
A sinusoidal tone is characterized by its A sinusoidal tone is characterized by its amplitude, frequency and phase.amplitude, frequency and phase.
Pure Tone Chord Musicnote_c5.wav chor_c5.wav beeth1.wav
Sinusoidal ToneSinusoidal Tone
Graph versus time for 10 ms:Graph versus time for 10 ms:
Formula: A*sin(2*pi*f*t+phi)Formula: A*sin(2*pi*f*t+phi)
note_c5.wav
Sinusoidal Tone: ParametersSinusoidal Tone: Parameters
In A*sin(2*pi*f*t+phi),In A*sin(2*pi*f*t+phi),AA is the is the amplitudeamplitude of the sinewave. A loud of the sinewave. A loud tone has a high amplitude.tone has a high amplitude.ff is the is the frequencyfrequency, i.e., the number of , i.e., the number of periodic repetitions per second. A low tone periodic repetitions per second. A low tone has low frequency, a high tone has high has low frequency, a high tone has high frequency. Note: pi = 3.14159265359...frequency. Note: pi = 3.14159265359...phiphi is the is the phasephase, i.e., the time shift of the , i.e., the time shift of the sinewave along the horizontal axis.sinewave along the horizontal axis.
Sinusoidal Tone: ParametersSinusoidal Tone: Parameters
The human ear is insensitive to the The human ear is insensitive to the phasephase of individual tones. Modems use of individual tones. Modems use their own reference signal to transmit or their own reference signal to transmit or receive data via phase changes.receive data via phase changes.AmplitudeAmplitude or loudness is straightforward.or loudness is straightforward.FrequencyFrequency is the most important is the most important parameter for us to characterize tones.parameter for us to characterize tones.
Sinusoidal Tone: ParametersSinusoidal Tone: Parameters
To understand more intuitively what To understand more intuitively what amplitude and frequency mean, let’s amplitude and frequency mean, let’s listen to an listen to an amplitude modulated (AM)amplitude modulated (AM)signal and to a signal and to a frequency modulated frequency modulated (FM)(FM) signal.signal.
No modulation AM FMnomodx.wav amx.wav fmx.wav
Sinusoidal Tone: FrequencySinusoidal Tone: Frequency
Example: Listen to this simple tune: Example: Listen to this simple tune: We recognize the melody because of We recognize the melody because of the specific changes in tone frequency.the specific changes in tone frequency.The sine function is periodic, i.e.,The sine function is periodic, i.e.,sin(x+2*pi) = sin(x)sin(x+2*pi) = sin(x), (2*pi is 360 degrees)., (2*pi is 360 degrees).
Frequency is defined as the number of Frequency is defined as the number of periods per second or, equivalently, as periods per second or, equivalently, as “one over the period length in seconds.”“one over the period length in seconds.”
twink.wav
Frequency: ExampleFrequency: ExampleThis graph shows 10 ms of a tone:This graph shows 10 ms of a tone:
What is the frequency of this tone?What is the frequency of this tone?
note_c5.wav
Frequency: ExampleFrequency: ExampleThis graph shows 10 ms of a tone:This graph shows 10 ms of a tone:
There are 5.25 periods. Thus, the There are 5.25 periods. Thus, the frequency is: frequency is: f = 100*5.25 = 525 Hzf = 100*5.25 = 525 Hz..
note_c5.wav
Frequency of TonesFrequency of Tones
Note that frequency is usually measured Note that frequency is usually measured in in HertzHertz (in honor of Heinrich Hertz, (in honor of Heinrich Hertz, 18571857--1894, German physicist who 1894, German physicist who discovered radio waves).discovered radio waves).Note:Note:
1,000 Hz = 1 kHz (kilo Hertz),1,000 Hz = 1 kHz (kilo Hertz),1,000,000 Hz = 1 MHz (mega Hertz),1,000,000 Hz = 1 MHz (mega Hertz),
1,000,000,000 Hz = 1 GHz (giga Hertz).1,000,000,000 Hz = 1 GHz (giga Hertz).
FrequenciesFrequencies
We can hear tones in the range from We can hear tones in the range from about 50 Hz to 16 kHz.about 50 Hz to 16 kHz.
AM radio uses 540 kHz … 1.6 MHz.AM radio uses 540 kHz … 1.6 MHz.FM radio uses 88 MHz … 108 MHz.FM radio uses 88 MHz … 108 MHz.Microwave ovens use 2 … 3 GHz.Microwave ovens use 2 … 3 GHz.
400 Hz 1000 Hz 4000 Hzsine_400.wav sin_1000.wav sin_4000.wav
Frequency DomainFrequency Domain
When we look at a plot of sound intensity When we look at a plot of sound intensity versus time, we say that we look at a versus time, we say that we look at a waveform in the waveform in the time domaintime domain..We can look at the same waveform We can look at the same waveform through filters which pass only certain through filters which pass only certain frequency bands, and plot intensity frequency bands, and plot intensity versus frequency. In this case we look at versus frequency. In this case we look at the waveform in the the waveform in the frequency domain.frequency domain.
Frequency Domain: ExampleFrequency Domain: Example
Spectrogram of 3 sequential tones:Spectrogram of 3 sequential tones:
sin_up.wav
Frequency Domain: ExampleFrequency Domain: Example
Spectrogram of chirp: 100 Hz … 20 kHzSpectrogram of chirp: 100 Hz … 20 kHz
chirp.wav
Frequency Domain: ExampleFrequency Domain: Example
Spectrogram of musical tune:Spectrogram of musical tune:
twink.wav
HarmonicsHarmonics
Let f1 be the fundamental frequency of Let f1 be the fundamental frequency of a sound, e.g., from a music instrument.a sound, e.g., from a music instrument.Then f2=2*f1, f3=3*f1, f4=4*f1, etc, are Then f2=2*f1, f3=3*f1, f4=4*f1, etc, are called the 2’nd, 3’rd, 4’th, etc, called the 2’nd, 3’rd, 4’th, etc, harmonicsharmonics..The presence (or absence) of The presence (or absence) of harmonics changes the “quality”, but not harmonics changes the “quality”, but not the “pitch” of the sound.the “pitch” of the sound.
Rectangular WaveformRectangular WaveformHas 1’st, 3’rd, 5’th, 7’th, etc, harmonics.Has 1’st, 3’rd, 5’th, 7’th, etc, harmonics.
4 periods in 10 ms => f1 = 400 Hz
rect_400
sin_400sin400_sig.wav
rect_sig.wav
Rectangular WaveformRectangular Waveform
400 Hz
1200 Hz
2000 Hz2800 Hz
3600 Hz
4400 Hz
5200 Hz
Sawtooth Sawtooth WaveformWaveformHas 1’st, 2’nd, 3’rd, 4’th, etc, harmonics.Has 1’st, 2’nd, 3’rd, 4’th, etc, harmonics.
4 periods in 10 ms => f1 = 400 Hz
saw_400
sin_400
rect_400
sin400_sig.wav
rect_sig.wav
saw_sig.wav
Sawtooth Sawtooth WaveformWaveform
400 Hz
1200 Hz
2000 Hz
2800 Hz2400 Hz
1600 Hz
800 Hz
Human VoiceHuman Voice
Most energy is concentrated in the band Most energy is concentrated in the band from about 500 to 2000 Hz.from about 500 to 2000 Hz.The telephone transmits frequencies in The telephone transmits frequencies in the range from 300 to 3000 Hz.the range from 300 to 3000 Hz.This is adequate for understanding This is adequate for understanding speech, but not for Hispeech, but not for Hi--Fi reproduction.Fi reproduction.
Hi-Fi Telephone(low pass)
High pass
infa_tel.wavinfa_ori.wav infa_hpf.wav
Human Voice: Time DomainHuman Voice: Time Domain
The HiThe Hi--Fi version versus time:Fi version versus time:
Speech is easily identified by the Speech is easily identified by the pauses between words and sentences.pauses between words and sentences.
infa_ori.wav
Human Voice: Frequency DomainHuman Voice: Frequency Domain
Spectrogram of HiSpectrogram of Hi--Fi version:Fi version:
Note: Most energy is below 4000 Hz.Note: Most energy is below 4000 Hz.
4 kHz
10 kHzinfa_ori.wav
Human Voice: Frequency DomainHuman Voice: Frequency Domain
Spectrogram of telephone version:Spectrogram of telephone version:
Now frequencies above 2.5 kHz and Now frequencies above 2.5 kHz and below 250 Hz are cut off.below 250 Hz are cut off.
4 kHz
infa_tel.wav
MusicMusicMusic typically covers a larger Music typically covers a larger frequency range than human voice.frequency range than human voice.More energy at lower frequencies, but More energy at lower frequencies, but higher frequencies are needed to higher frequencies are needed to distinguish different instruments.distinguish different instruments.For HiFor Hi--Fi quality, 50…16000 Hz range is Fi quality, 50…16000 Hz range is needed.needed.
50..20000 Hz 50..10000 Hz 50..5000 Hz
muss44.wav muss22.wav muss11.wav
Music: Time DomainMusic: Time DomainThe 50..20000 Hz version versus time:The 50..20000 Hz version versus time:
Intensity fluctuates in time, but pauses Intensity fluctuates in time, but pauses are much rarer than with speech.are much rarer than with speech.
muss44.wav
Music: Frequency DomainMusic: Frequency Domain
Spectrogram of 50..20000 Hz version:Spectrogram of 50..20000 Hz version:
High frequency content substantial.High frequency content substantial.
10 kHz
20 kHzmuss44.wav
Music: Frequency DomainMusic: Frequency Domain
Spectrogram of 50..5000 Hz version:Spectrogram of 50..5000 Hz version:
Same tune, but “less HiSame tune, but “less Hi--Fi” sound.Fi” sound.
10 kHz
20 kHzmuss11.wav
DTMF SignalsDTMF Signals
Dual Tone MultiDual Tone Multi--Frequency signals are Frequency signals are used extensively in telephony.used extensively in telephony.
1209 Hz 1336 Hz 1477 Hz
697 Hz 1 2 3
770 Hz 4 5 6
852 Hz 7 8 9
941 Hz * 0 #
Example
dtmf_x.wav
DTMF Signal: Time DomainDTMF Signal: Time Domain
How do we know frequencies?How do we know frequencies?
Do we even know there are two?Do we even know there are two?
dtmf_x.wav
DTMF Signal: SpectrogramDTMF Signal: Spectrogram
Use filters (frequency domain).Use filters (frequency domain).
We see that there are 2 frequencies.We see that there are 2 frequencies.
dtmf_x.wav
Different Types of FiltersDifferent Types of Filters
LPF:LPF: Lowpass Lowpass filter, passes all filter, passes all frequencies below a cutoff frequency.frequencies below a cutoff frequency.BPF:BPF: Bandpass Bandpass filter, passes all filter, passes all frequencies from a lower to an upper frequencies from a lower to an upper cutoff frequency.cutoff frequency.HPF:HPF: Highpass Highpass filter, passes all filter, passes all frequencies above a cutoff frequency.frequencies above a cutoff frequency.
DTMF Signal: After LPFDTMF Signal: After LPF
After lowAfter low--pass filter (LPF) at 1000 Hz:pass filter (LPF) at 1000 Hz:
7.75 periods in 10 ms ==> f=775 Hz
dtmf_x_lpf.wav
DTMF Signal: After HPFDTMF Signal: After HPF
After highAfter high--pass filter (HPF) at 1000 Hz:pass filter (HPF) at 1000 Hz:
14.75 periods in 10 ms ==> f=1475 Hz
dtmf_x_hpf.wav
White NoiseWhite Noise
Contains all frequencies with equal power.Contains all frequencies with equal power.
Random signal, no visible regularity
wnoise_sig.wav
Pink NoisePink Noise
Equal power per octave.Equal power per octave.
pink
whitewnoise_sig.wav
pink1.wav
Spectrogram of Pink NoiseSpectrogram of Pink Noise