Analog and Digital Signals

Transcription

1 E.M. Bakker LML Audio Processing and Indexing 1 Analog and Digital Signals 1. From Analog to Digital Signal 2. Sampling & Aliasing LML Audio Processing and Indexing 2 1

2 Analog and Digital Signals Analog Signals Continuous function F of a continuous variable t (t can be time, space etc) : F(t) Digital Signals Discrete function F k of a discrete (sampling) variable t k, with k an integer: F k = F(t k). Uniform (periodic) sampling with sampling frequency f S = 1/ t S, e.g., t s = sec => f s = 1000Hz LML Audio Processing and Indexing 3 Digital System Implementation Analog Input Important issues: Analysis bandwidth, Dynamic range Antialiasing Filter Pass/stop bands A/D Conversion Digital Processing Sampling rate, Number of bits, and further parameters Digital format Digital Output LML Audio Processing and Indexing 4 2

3 Sampling How fast must we sample a continuous signal to preserve its information content? Examples: Turning wheels of a train in a movie 25 frames per second, i.e., 25 samples/sec. Train starts => wheels appear to go clockwise Train accelerates => wheels go counter clockwise Rotating propeller of an airplane captured by a Mobile phone camera. Frequency misidentification due to low sampling frequency. Sampling: independent variable (for example time) Quantisation: dependent variable (for example voltage) continuous -> discrete continuous -> discrete. Here we will talk about uniform sampling. LML Audio Processing and Indexing 5 Sampling t s(t) = sin(2f0t) f S For example: f 0 = 1 Hz, f S = 3 Hz s1 (t) = sin(24t) s2 (t) = sin(27t) f S represents exactly all sine-waves s k (t) defined by: s k (t) = sin( 2 (f 0 + k f S ) t ), k N LML Audio Processing and Indexing 6 3

4 The sampling theorem Theorem A signal s(t) with maximum frequency f MAX can be recovered if sampled at frequency f S > 2 f MAX. * Multiple proposers: Whittaker(s), Nyquist, Shannon, Kotel nikov. Nyquist frequency (rate) f N = 2 f MAX Example s(t) 3cos(50πt) 10sin(300πt) cos(100πt) Condition on f S? F 1 F 2 F 3 F 1 =25 Hz, F 2 = 150 Hz, F 3 = 50 Hz f S > 300 Hz f MAX LML Audio Processing and Indexing 7 Frequency Domain Time and Frequency are two complementary signal descriptions. Signal can be seen as projected onto the time domain or frequency domain. Bandwidth indicates the width of a range in the frequency domain. The range can be located high up in the frequency domain, then we talk about high bandwidth. Passband Bandwidth => lower and upper cutoff frequencies As we have seen from the previous lecture act the inner-ear together with early neural circuitry as a frequency analyser. The audio spectrum is split into narrow bands thereby enabling detection of low-power sounds out of louder background sounds. LML Audio Processing and Indexing 8 4

5 Sampling Low-Pass Signals (a) Continuous spectrum (a) Band-limited signal: frequencies of the signal assumed in [-B, B] (f MAX = B). -B 0 B f (b) Discrete spectrum No aliasing (b) Time sampling frequency repetition. f S > 2 B no aliasing. (c) -B 0 B f S/2 f Discrete spectrum Aliasing & corruption Note: s(t) at f S represents all sine-waves sk(t) defined by: sk (t) = sin( 2 (f 0 + k f S ) t ), k N (c) f S 2 B aliasing! 0 f S/2 f Aliasing: signal ambiguity in frequency domain f S LML Audio Processing and Indexing 9 Sampling Low-Pass Signals Discrete spectrum Aliasing & corruption 0 fs/2 f - f S - f S /2 0 f S /2 B f S f S 2 B aliasing! Aliasing: signal ambiguity in frequency domain Signal For example out of band noise. LML Audio Processing and Indexing 10 5

6 Antialiasing Filter (a) Out of band noise Signal of interest Out of band noise (a),(b) Out-of-band noise can aliase into band of interest. Filter it before! (b) -B 0 B f Out of band noise(t) will be sampled: f S thereby mimicking a non-existing frequency within the band. (c) -B 0 B f S/2 f (c) Antialiasing filter Passband: depends on bandwidth of interest. LML Audio Processing and Indexing 11 Under-sampling Using spectral replications to reduce sampling frequency f S requirements. Bandpass signal centered on f C B 2f C B 2f C B f S m1 m 0 f C f mn, selected so that f S > 2B Example f C = 20 MHz, B = 5MHz Without under-sampling f S > 40 MHz. With under-sampling: f S = 22.5 MHz (m=1) f S = 17.5 MHz (m=2) f S = MHz (m=3) -f S 0 f S 2f S f >2B Advantages Slower ADCs / electronics needed. Simpler antialiasing filters. f C LML Audio Processing and Indexing 12 6

7 Over-sampling Oversampling : sampling at frequencies f S >> 2 f MAX. Over-sampling & averaging may improve ADC resolution f OS = 4 w f S f OS = over-sampling frequency, w = additional bits required. Each additional bit implies/requires over-sampling by a factor of four. LML Audio Processing and Indexing 13 (Some) ADC parameters 1. Number of bits N (~resolution) 2. Data throughput (~speed) 3. Signal-to-noise ratio (SNR) 4. Signal-to-noise-&-distortion rate (SINAD) 5. Effective Number of Bits (ENOB) 6. Different applications have different needs. Radar systems Static distortion Communication Imaging / video NB: Definitions may be slightly manufacturer-dependent! LML Audio Processing and Indexing 14 7

8 ADC - Number of bits N Continuous input signal digitized into 2 N levels signal Uniform, bipolar transfer function (number of bits N=3 => 8 levels) V Quantisation step q = Ex: V max = 1V, N = 12 V max 2 N q = V V FSR q / 2 Voltage ( = q) Scale factor (= 1 / 2 N ) Percentage (= 100 / 2 N ) q / 2-1 Quantisation error LML Audio Processing and Indexing 15 ADC - Quantisation error Quantisation step q = V max 2 N Voltage [V] time [ms] Quantisation Error e q in [-0.5 q, +0.5 q]. e q limits ability to resolve small signal. Higher resolution means lower e q. e q [V] 10-4 QE for N = 12 V FS = Sampling time, t k LML Audio Processing and Indexing 16 8

9 SNR of ideal ADC RMS input SNR ideal 20log 10 (1) RMS(e q ) Also called SQNR (signal-to-quantisation-noise ratio) (RMS = root mean square) RMS T 1 V FSR T 2 0 input sin ωt 2 V dt FSR 2 2 Assumptions Ideal ADC: only quantisation error e q ( p(e) = quantisation error probability density is assumed to be constant, uniform, etc. ) e q uncorrelated with signal. ADC performance constant in time. Input(t) = ½ V FSR sin( t). p(e) quantisation error probability density RMS(eq) q/2 2 eq peq deq -q/2 q VFSR 12 2 N 12 1 q (sampling frequency f S = 2 f MAX ) q 2 q 2 e q Error value LML Audio Processing and Indexing 17 SNR of ideal ADC Substituting in (1) => SNR ideal 6.02N1.76 [db] (2) One additional bit SNR increased by 6 db Real SNR lower because: - Real signals have noise. - Forcing input to full scale unwise. - Real ADCs have additional noise (aperture jitter, non-linearities etc). Actually (2) needs correction factor depending on ratio between sampling freq & Nyquist freq. Processing gain due to oversampling. LML Audio Processing and Indexing 18 9

10 ADC Performance Currently: ~3 bits higher From: LML Audio Processing and Indexing 19 Finite word-length effects Overflow : arises when arithmetic operation result has one too many bits to be represented in a certain format. Dynamic range db = 20 log 10 largest value smallest value Fixed point ~ 180 db Floating point ~1500 db High dynamic range => wide data set representation with no overflow. Note: Different applications have different needs. For example: Telecommunication: 50 db HiFi audio: 90 db. LML Audio Processing and Indexing 20 10

11 Complex Numbers The complex numbers are given by: C = c c = a + bi, where, a, b R} here i is the imaginary unit that satisfies: i 2 = 1 a is called the real part of c b is called the imaginary part of c If z=x+yi, then the complex conjugate z * is defined as z * =x-yi LML Audio Processing and Indexing 21 Complex Numbers (see also Wikipedia) The complex numbers are given by: C = c c = a + bi, where, a, b R} here is the imaginary unit that satisfies: i 2 = 1 Addition: a + bi + c + di = a + c + b + d i a + bi c + di = a c + b d)i Multiplication: a + bi c + di = ac bd + (bc + ad)i a + bi (a + bi)(c di) = c + di (c + di)(c di) = ab + bd bc ad c 2 + d 2 + c 2 + d 2 i LML Audio Processing and Indexing 22 11

12 Complex Numbers The complex numbers are given by: C = c c = a + bi, where, a, b R} The absolute value (modulus; magnitude) of z = x + yi is: r = z = x 2 + y 2 Note that: z 2 = zz = x 2 + y 2 The argument (phase) of z = x + yi is: φ = arg z = {arctan(y/x), if = "the angle of the vector (x,y) with the positive real axis Note: z = rcosφ + isinφ = re iφ LML Audio Processing and Indexing 23 Complex Numbers Let: Note: z 1 = r 1 cosφ 1 + isinφ 1 = r 1 e iφ 1 z 2 = r 2 cosφ 2 + isinφ 2 = r 2 e iφ 2 cos a cos b sin a sin b cos a sin b + sin a cos b = cos a + b = sin(a + b) Hence: z 1 z 2 = r 1 r 2 cos(φ 1 +φ 2 + isin(φ 1 +φ 2 )) = r 1 r 2 e i(φ 1+φ 2 ) LML Audio Processing and Indexing 24 12

13 Sine Cosine Graphs sin φ + π/2 = cos(φ) LML Audio Processing and Indexing 25 References This presentation uses a selection of slides that are adapted from original slides by Dr M.E. Angoletta at DISP2003, a DSP course given by CERN and University of Lausanne (UNIL) LML Audio Processing and Indexing 26 13