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Analog Signals Continuous-Time Signals Discrete-Time, Digital Signals Analog Signals Music 171: Fundamentals of Digital Audio • Tamara Smyth, trsmyth@ucsd.edu A signal, of which a sinusoid is only one example, is Department of Music, a set, or sequence of numbers. University of California, San Diego (UCSD) • The term “analog” refers to the fact that it is October 15, 2019 “analogous” of the signal it represents. • A “real-world” signal is captured using a microphone which has a diaphragm that is pushed back and forth according to the compression and rarefaction of the sounding pressure waveform. Figure 1: The electrical signal used to represent the pressure variations of a sound wave is an analog signal. • The microphone transforms this displacement into a time-varying voltage—an analog electrical signal. 1 Music 171: Fundamentals of Digital Audio 2 Continuous-Time Signals Discrete-Time, Digital Signals • Analog signals are continuous in time. • When analog signals are brought into a computer, they must be made discrete (finite and countable). • A continuous-time signal is an infinite and uncountable sequence of numbers, as are the possible • A discrete-time signal is a finite sequence of values each number can have. numbers, with finite possible values for each number. – – between a start and end time, there are infinite number values are limited by how many bits are possible values for time t and the waveform’s used to represent them (bit depth); instantaneous amplitude x(t). – can be stored on a digital storage medium. • A continuous signal cannot be stored, or processed, in a computer since it would require infinite data. • Analog signals must be discretized (digitized) to produce a finite set of numbers for computer use. • Sampling: the process of taking individual values of a continuous-time signal (at regular time intervals). Music 171: Fundamentals of Digital Audio 3 Music 171: Fundamentals of Digital Audio 4 Analog to Digital Conversion Sampling • The process by which an analog signal is digitized is • Sampling: process of taking values (samples) of the called analog-to-digital or “a-to-d” conversion analog waveform at regularly spaced time intervals. • analog-to-digital converter (ADC): the device x(t) x[n] = x(nTs) that discretizes (digitizes) an analog signal. ADC • The ADC must accomplish two (2) tasks: 1. sampling: Ts = 1/fs – taking values (samples) at regular time intervals; Figure 2: The ideal analog-to-digital converter. 2. quantization: – assign a number to the value (using limited computer bits). • Sampling Rate: number of samples taken per second (Hz); – fs = 48 kHz (professional studio) – fs = 44.1 kHz (CD) – fs = 32 kHz (broadcasting) • Sampling Period: time interval (in seconds) between samples: Ts =1/fs seconds. Music 171: Fundamentals of Digital Audio 5 Music 171: Fundamentals of Digital Audio 6 Sampled Sinusoids Question 1 • Sampling corresponds to transforming the continuous • If the following sinusoid was sampled at fs = 16 Hz, time variable t into a set of discrete times that are – what is the duration of the signal shown? integer n multiples of the sampling period Ts: 1 t −→ nTs. 0.8 • Integer n corresponds to the index in the sequence. 0.6 • Continuous sinusoid: 0.4 0.2 x(t)= A sin(ωt + φ). 0 • Discrete sinusoid −0.2 −0.4 x(n)= A sin(ωnTs + φ), −0.6 a sequence of numbers that may be indexed by n. −0.8 −1 0 5 10 15 20 25 Sample Index • What would the duration be if fs = 32? Music 171: Fundamentals of Digital Audio 7 Music 171: Fundamentals of Digital Audio 8 • Answer 1 If fs = 32, the duration is shorter: 24 3 24 × T = = = .75 s. s 32 4 • If fs = 16, what is the duration shown? 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 0 5 10 15 20 25 Sample Index • The sampling period (time between samples) is Ts =1/16 s. Since 24 samples are shown, the duration is 24 3 24 × T = = =1.5 s. s 16 2 Music 171: Fundamentals of Digital Audio 9 Music 171: Fundamentals of Digital Audio 10 Question 2 Answer 2 (Method 1) • If the following sinusoid was sampled at fs = 16 Hz, • If fs = 16, what is the frequency of the sinusoid? – what is the frequency of the sinusoid? 1 0.8 1 0.6 0.8 0.4 0.6 0.2 0.4 0 0.2 −0.2 0 −0.4 −0.2 −0.6 −0.4 −0.8 −0.6 −1 −0.8 0 5 10 15 20 25 Sample Index −1 0 5 10 15 20 25 Sample Index • Method 1: – 16 samples corresponds to 1 second, – there are 2 cycles in 1 second (after 16 samples), – the frequency is 2 Hz. Music 171: Fundamentals of Digital Audio 11 Music 171: Fundamentals of Digital Audio 12 • Answer 2 (Method 2) If fs = 32, then the frequency is higher: 1 32 f = = =4 Hz. T 8 • If fs = 16, what is the frequency of the sinusoid? 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 0 5 10 15 20 25 Sample Index • Method 2: the period of the sinusoid is 8 1 T =8 × T = = , s 16 2 and the frequency is 1 1 f = = =2 Hz. T 2 Music 171: Fundamentals of Digital Audio 13 Music 171: Fundamentals of Digital Audio 14 Sampling and Reconstruction Importance of Knowing Sampling Rate • Once x(t) is sampled to produce x(n), time scale • If the signal is digitized and reconstructed using the information is lost. same sampling rate, the frequency and duration will • x(n) may represent a number of possible waveforms. be preserved. • If reconstruction is done using a different sampling Continuous 2-Hz Sinusoid 1 rate, the time interval between samples 0.5 changes resulting in a change in 0 -0.5 – overall signal duration, -1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 – time to complete one cycle (period) and thus Time (s) 2-Hz Sinusoid Sampled at 32 Hz 1 sounding frequency. 0.5 0 -0.5 -1 0 5 10 15 20 25 30 Sample index 2-Hz Sinusoid Sampled at 32 Hz and Reconstructed at 16 Hz 1 0.5 0 -0.5 -1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time (s) • Reconstructing at half the sampling rate (Fs/2) will double the time between samples (2/Fs), making the sinusoid twice as long and halving the frequency. Music 171: Fundamentals of Digital Audio 15 Music 171: Fundamentals of Digital Audio 16 Question 3 Answer 3 • A 220-Hz sinusoid is sampled at fs1 = 44100. It is • then played on an audio system having a different If a 220-Hz sinusoid sampled at fs1 = 44100 Hz is played back at fs2 = 22050 Hz (fs1/2), then the sampling rate of fs2 = 22050 Hz. – – At what frequency will the sinusoid sound? period between the samples will be twice as long: 1 1 2 2 Ts2 = = = = , fs2 22050 44100 fs1 – and the period of oscillation will double: 2 2 T2 =2T1 = = s, f1 220 – and the corresponding frequency will be halved (sound an octave lower): 1 220 f2 = = = 110 Hz. T2 2 Music 171: Fundamentals of Digital Audio 17 Music 171: Fundamentals of Digital Audio 18 Sampling in Practice II Question 4 • In the Beatles track “In My Life” (1:28) there is a • What is the sampling rate and frequency of the Baroque-style piano solo composed and played by following sinusoid? George Martin: 1 – piano solo 0.8 Problem: George Martin’s couldn’t play fast enough! 0.6 Solution: 0.4 0.2 – play an octave lower and at half tempo 0 – record at 1/2 sampling rate (or “speed” if analog), −0.2 – play back at twice the rate at which it was −0.4 recorded. −0.6 Effect: The distortion creates a sort of harpsichord −0.8 sound at the regular tempo and pitch. Why? −1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 • Listen to how the original sound regains a piano Time (s) quality when played at 1/2 the sampling rate: – piano solo at 1/2 sampling rate • Note: Beatles recordings were actually analog, but the same principles apply! Music 171: Fundamentals of Digital Audio 19 Music 171: Fundamentals of Digital Audio 20 Answer 4 Implications of Sampling • Is a sampled sequence only an approximation of the • Give the sinusoid’s sampling rate and frequency. original? 1 • perfectly 0.8 Is it possible to reconstruct a sampled signal? 0.6 • 0.4 Will anything less than an infinite sampling rate 0.2 introduce error? 0 • How frequently must we sample in order to −0.2 “faithfully” reproduce an analog waveform? −0.4 −0.6 −0.8 −1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Time (s) • The period is 2 seconds (or, there is 1/2 cycle after 1 second) and the frequency is 1/2 Hz. • The sinusoid has 8 samples in 1 second and thus the sampling rate is fs =8 Hz. • The period has 16 samples and is thus 16 × 1/8=2 seconds long. The frequency is thus .5 Hz. Music 171: Fundamentals of Digital Audio 21 Music 171: Fundamentals of Digital Audio 22 Nyquist Sampling Theorem Digital Audio System • The Nyquist Sampling Theorem states that: A bandlimited continuous-time signal can be x(t) low−pass low−pass x(nTs) filter ADCCOMPUTER DAC filter sampled and perfectly reconstructed from its samples if the waveform is sampled over twice as fast as it’s highest frequency component.
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