Complex Quadrature Signals in Biomedical Signal Processing

Mesleki ngilizce - Technical English II Prof. Dr. Nizamettin AYDIN naydin@yildiz.edu.tr http://www.yildiz.edu.tr/~naydin 1

Notes: In the slides, texts enclosed by curly parenthesis, { }, are examples. texts enclosed by square parenthesis, [ ], are explanations related to examples. 2

Processing Complex Quadrature Signals Learning Objectives to understand inter-disciplinary nature of biomedical signal processing to become familiar with ultrasound applications to understanding, the limits, capabilities, and benefits of digital signal processing Sub-areas covered mathematics electronics transform domain processing algorithms 3

Processing Complex Quadrature Signals Keywords Complex signals a two-dimensional signal whose value at some instant in time can be specified as a single complex number having two parts: real part and imaginary part Quadrature Signals signals that based on the notion of complex numbers used in many digital signal processing applications same as the complex signals Doppler effect/shift The Doppler effect (or the Doppler shift) is the change in frequency or wavelength of a wave for an observer who is moving relative to the wave source 4

Processing Complex Quadrature Signals Keywords ultrasound sound at frequencies greater than 20 kHz demodulator an electronic circuit (or computer program in a software- defined radio) that is used to recover the information content from the modulated carrier wave time frequency analysis time frequency analysis comprises those techniques that study a signal in both the time and frequency domains simultaneously, using various time frequency representations. 5

Processing Complex Quadrature Signals Keywords Fourier transform The Fourier transform (FT) decomposes a function of time (a signal) into the frequencies that make it up, in a way similar to how a musical chord can be expressed as the frequencies (or pitches) of its constituent notes sonogram a graph representing a sound, showing the distribution of energy at different frequencies a visual image produced from an ultrasound examination wavelet A wavelet is a mathematical function used to divide a given function or continuous-time signal into different scale components. Usually one can assign a frequency range to each scale component 6

Processing Complex Quadrature Signals Reading text Pre-reading questions ? 7

Physics and Principles of Ultrasound Ultrasound has a storied history which achieved reality during World War II in finding German submarines in the protection of North Atlantic convoys. It was initially employed for its therapeutic benefits in physical therapy to produce deep heat and ablation of brain lesions for Parkinson s disease. The wave characteristics have been altered in diagnostic ultrasound to the point where energy transfer is limited, and deep tissue heat is virtually nonexistent. 8

Physics and Principles of Ultrasound Ultrasound systems are comprised of a transducer, console (which contains the computer software, electrical components, Doppler technology, and storage), and the display. The physics of ultrasound waves and the means of their delivery are important to meld into this discussion. Some of the creatures use sound waves in a remarkable way. Sound waves depend on a support transport medium 9

Physics and Principles of Ultrasound Dolphins and odontocetes such as toothed whales emit very-high-pitched single-frequency clicks to communicate with others of their species It is also used for echo localization of schools of fish upon which they prey. Baleen whales which feed on plankton do not transmit in the high-frequency range but more on the order of 10 30 Hz. These sound waves travel extreme distances due to both their low frequency and the medium of transport which has important communication advantages. 10

Physics and Principles of Ultrasound The principles of ultrasound are complex, relying on sophisticated physics and mathematics. Sound is transmitted as sequential sine waves whose height represents amplitude or loudness 11

Physics and Principles of Ultrasound The human ear can recognize sounds as low as 20 Hz and as high as 20,000 Hz, and ultrasound is so named because its frequency emission is in the range of more than a million cycles per second or in the megahertz range. 12

Physics and Principles of Ultrasound An ultrasound wave is transmitted to human tissues through the transducer by physical deformation of the tissue surface. This is accomplished through piezoelectric crystals which elongate and shorten in response to applied alternating electrical current 13

Physics and Principles of Ultrasound Elephants also transmit in this approximate low- frequency range with a volume level which may reach 117 dB. These transmissions can be identified as far as 10 km from the source and may also be sensed by the broad elephant s feet or the trunk which it may place along the ground to hear in this unique way. Bats emit sounds in the ultrasound range to identify insects and obstructions but the waves are disadvantaged by having to travel in air 14

Doppler Japanese scientist Satomura in 1956 observed that erythrocytes can reflect ultrasonic waves and this effect is called the Doppler effect. Doppler effect is a change in frequency of the reflected ultrasound waves backscattered from the structure that is in motion. 15

Doppler ultrasound As an object emitting sound moves at a velocity v, the wavelength of the sound in the forward direction is compressed ( s) and the wavelength of the sound in the receding direction is elongated ( l). Since frequency (f) is inversely related to wavelength, the compression increases the perceived frequency and the elongation decreases the perceived frequency. c = sound speed. Doppler tutorial 16

Doppler ultrasound cos vf = =2 = = f f f t d t r c ftis transmitted frequency fr is received frequency v is the velocity of the target, isthe angle between the ultrasound beam and the direction of the target's motion, and c is the velocity of sound in the medium f c d cos = =2 v f t 17

A general Doppler ultrasound signal measurement system audio-visual display store print etc. acoustical energy electrical energy out Transmission & Reception Electrical Display & Further processing in Processing 18

Processing of Doppler Ultrasound Signals sin cos si Gated transmiter Master osc. Sample & hold Band-pass filter Demodulator Further processing Logic unit sq Receiver amplifier RF filter Sample & hold Band-pass filter Demodulator Transducer Demodulation Quadrature to directional signal conversion Time-frequency/scale analysis Data visualization Detection and estimation Derivation of diagnostic information V 19

Quadrature phase detection 0 + f 0f 0f yD LPF 0 cos t RF signal Quad. signal oscillator 0 sin t yQ LPF 20

Quadrature Signals Quadrature signals are based on the notion of complex numbers used in many digital signal processing applications such as Communication Radar Sonar Ultrasound MR imaging Direction finding schemes Antenna beamforming applications Single sideband modulators 21

A quadrature signal is a two dimensional signal whose value at any given time can be specified by a single complex number Such as a(t)+jb(t) Quadrature signal processing is used in many fields of science and engineering Processing of complex quadrature signal provides additional processing power by enabling to measure amplitude and phase of a signal simultaneously 22

General Definition of Quadrature Doppler Signal A general definition of a discrete quadrature Doppler signal equation can be given by = ( n ( ) ( ) [ ( )] n D n s n H s n f r = ( ) [ )] ( ) Q n H s s f r D(n) and Q(n), each containing information concerning forward channel and reverse channel signals (sf(n) and sr(n) and their Hilbert transforms H[sf(n)] and H[sr(n)]), are real signals. 23

Properties of Complex Quadrature Doppler US signals = + = ( ) , [ ] s t D jQ D H Q + R R + I I = + + + ( ) ( ) ( ) ( ( ) ( )) S S S j S S 2 ( ) 0 0 0 S if if = = ( ) , ( ) S S f r 0 0 2 ( ) 0 if S if + + = + + + = + ( ) { ( ) ( )} { ( ) ( )} ( ) ( ) S S jS S jS S S R I R I f r 24

DSP for Quadrature to Directional Signal Conversion Time domain methods Phasing filter technique (PFT) (time domain Hilbert transform) Weaver receiver technique Frequency domain methods Frequency domain Hilbert transform Complex FFT Spectral translocation Scale domain methods Complex continuous wavelet Complex discrete wavelets 25

Hilbert Transform... The Hilbert transform (HT) is a widely used frequency domain transform. It shifts the phase of positive frequency components by -900and negative frequency components by +900. The HT of a given function x(t) is defined by the convolution between this function and the impulse response of the HT (1/ t). 1 1 ( ) x + = = d [ ( )] ( ) H x t x t t t 26

...Hilbert Transform Specifically, if X(f) is the Fourier transform of x(t), its Hilbert transform is represented by XH(f), where = = = ( ) [ ( )] ( ) ( ) ( sgn ) ( ) X f H X f H f X f j f X f H H A 900phase shift is equivalent to multiplying by ej900= j, so the transfer function of the HT HH(f) can be written as , 0 j f = = ( ) sgn HH f j f + , 0 j f 27

Impulse Response of HT = ) 2 / , 0 2 0 n 2h (n) H 1 = ( ) hH n 2 sin ( n , 0 n n 1/3 1/5 -2 -4 -6 -3 -1 -5 An ideal HT filter can be approximated using standard filter design techniques. If a FIR filter is to be used , only a finite number of samples of the impulse response suggested in the figure would be utilised. 3 1 5 6 4 n 2 -1/5 -1/3 -1 28

Complex Modulation j t ( ) ( ) x t e X c c X(f) X(f-f ) c f f 0 0 -W W f -W c f +W c f c = j + + {cos } { ( ) ( )} F t 0 0 0 = + {sin } { ( ) ( )} F t 0 0 0 29

x(t)ejtis not a real time function and cannot occur as a communication signal. However, signals of the form x(t)cos( t+ ) are common and the related modulation theorem can be given as j X t t x + j e e + + ( ) cos( ) ( ) ( ) X c c c 2 2 So, multiplying a band limited signal by a sinusoidal signal translates its spectrum up and down in frequency by c 30

Asymmetrical implementation of the PFT DSP Algorithm Doppler signal equation: D(n) HILBERT TRANSFORM + y (n) f + = ( n ( ) ( ) [ ( )] n D n s n H s n f r = ( ) [ )] ( ) Q n H s s f r - Mathematical description of algoritm: = Q(n) + y (n) r DELAY FILTER = [ ( )] [ ( ) [ ( )]] [ ( )] ( ) H D n H s n H s n H s n s n f r f r = + = ( ) ( ) [ ( )] [ ( )] ( ) [ ( )] ( ) y n Q n H D n H s n s n H s n s n f f r f r = = ( ) ( ) [ ( )] [ ( )] ( ) [ ( )] ( ) y n Q n H D n H s n s n H s n s n r f r f r = ( ) 2 [ ( )] y n H s n f f = ( ) 2 ( ) y n s n r r 31

Symmetrical implementation of the PFT DSP Algorithm y (n) f D(n) HILBERT TRANS. = + ( n ( ) ( s ) [ ( )] n D n s n H s n f r DELAY FILTER = + ( ) [ )] ( ) Q n H s f r DELAY FILTER y (n) r Q(n) HILBERT TRANS. = = + + = = H D n [ ( )] H s n H s n [ ( )]] H s n s n r ( ) [ ( ) [ ( )] f r f = = + + = = + + H Q n [ ( )] H H s [ [ n s n r ( )] s n H s n [ ( )] ( )] ( ) f f r = + = ( ) ( ) [ ( )] 2 [ ( )] = = y n Q n H D n H s n ( ) ( ) [ ( )] 2 ( ) y n D n H Q n s n f f f f = + = ( ) ( ) [ ( )] 2 [ ( )] y n D n H Q n H s n = = ( ) ( ) [ ( )] 2 ( ) y n Q n H D n s n r r r r 32

Simulation result TIME DOMAIN WAVEFORMS 2000 D -2000 2000 Q -2000 2000 D_H -2000 2000 Q_D -2000 2000 OA -2000 2000 OB -2000 0 2 6 10 12 14 16 8 4 (Time, 1 ms/Div) OUTPUT SPECTRA 2000 0 0 1 2 3 4 5 6 7 8 8 7 6 5 4 3 2 1 (Frequency, 1 kHz/Div) 33

DSP Algorithm An alternative algorithm is to implement the HT using phase splitting networks A phase splitter is an all-pass filter which produces a quadrature signal pair from a single input The main advantage of this algorithm over the single filter HT is that the two filters have almost identical pass- band ripple characteristics f(n) y +450 D(n) Phase splitter (900) -450 -450 Q(n) Phase splitter (900) r(n) y +450 34

Weaver Receiver Technique (WRT) For a theoretical description of the system consider the quadrature Doppler signal defined by = + ( n ( ) ( s ) [ ( )] n D n s n H s n f r = + ( ) [ )] ( ) Q n H s f r which is band limited to fs/4, and a pair of quadrature pilot frequency signals given by p n n p n d c ( ) sin , = = = = n ( ) cos q c where c/2 =fs/4. The LPF is assumed to be an ideal LPF having a cut-off frequency of fs/4. 35

Asymmetrical implementation of the WRT DSP Algorithm p d (n) f(n) D(n) y X1 X2 + X3 LPF LPF = + ( n + ( ) ( s ) [ ( )] n D n s n H s n f r = + ( ) [ )] ( ) Q n H s f r + Y3 r(n) y Q(n) Y2 - Y1 LPF LPF q(n) p fp=fc=fs/4 = = = = p n d ( ) n p n , n sin ( ) cos c q c 36

= = X Y D n p n ( ). Q n p n ( ). 2 , 2 ( ) ( ) d q = = + + } { [ n + + s ( ).sin n n [ ( )].sin H s n r H s ( )].cos n n ( )cos s n r n { } f c c f c c ( ), 0 jS f = = ( ) = = ( ) F s { n S and F s n S ( )} { ( )} = = { [ ( )]} [ ( )] F H s n H S f f + f f r r ( ), 0 jS f + f ( ), 0 jS = ( ) ( ), 0 S S r = = { [ ( )]} [ ( )] F H s n H S f r r + ( ), 0 jS f = r ( ) ( ), 0 S S f + r = ( ) ( ), 0 S S r r = ( ) ( ), 0 S S r + f f + f f = + = + ( ) ( ) ( ) [ ( )] ( ) ( ) S S S H S jS jS f f + r r + r r = + = + ( ) ( ) ( ) [ ( )] ( ) ( ) S S S H S jS jS r r 37

= = ( + + ( + + H S [ S )] ) 2 = = ( + + ( + + )} { + + ( + + + + ( + + + + F X { jS jS S S } { ) ) )} f c r c f c f c r c r c = = ( + + ( H S [ S )] ) 2 = = ( + + ( )} { + + ( ( + + + + + + F Y { jS jS S S } { ) ) )} f c r c f c f c r c r c 2 = = ( = = ( + + ( + + + + F X { H S [ jS jS } )] ) ) f c f c S f c 2 = = ( = = ( ( + + + + F Y { S S } ) ) ) r c r c r c 1 2 1 2 1 2 1 2 1 2 1 2 = = ( ) + + ( ) ( 2 ( + + 2 = = ( ) 2 + + + + F X { S S S S S S 3 } ) ) ( ) f f f c f c f f c 1 2 1 2 1 2 1 2 1 2 1 2 = = ( ) ( ) ( 2 ( + + 2 = = ( ) 2 + + + + F Y { S S S S S S 3 } ) ) ( ) r r r c r c r r c 1 1 1 = { ( )} ( ), F y n S 1 = = ( ) { ( )} ( ), y n F S s n f f 2 f f f 2 2 1 1 1 = { ( )} ( ). F y n S 1 = = ( ) { ( )} ( ). y n F S s n r r 2 r r r 2 2 38

Graphical description D 1 Q 2 pd 3 pq 4 X1 5 + + + + + + t t j t j t sin cos cos sin f r f r Y1 6 X2 7 Y2 8 yf 9 yr 10 -2fc -fc -f 0 +f +fr +fc +2fc -fr f f Lowpass filter cut-off frequency=f , f =f /4 c c s Stop-band region Pass-band region 39

Symmetrical implementation DSP Algorithm f(n) y A1 A2 LPF LPF D(n) d(n) p q(n) p Q(n) r(n) y B1 B2 LPF LPF 40

Simulation result TIME DOMAIN WAVEFORMS 2000 D -2000 2000 Q -2000 2000 X1 -2000 2000 Y1 -2000 2000 X2 -2000 0 2 4 6 8 10 12 14 16 (Time, 1 ms/Div) OUTPUT SPECTRUM 1000 0 0(4) 1(3) 2(2) 3(1) 4(0) 5(1) 6(2) 7(3) 8(4) (Frequency, 0.5 kHz/Div) 41

FREQUENCY DOMAIN PROCESSING These algorithms are almost entirely implemented in the frequency domain (after fast Fourier transform), They are based on the complex FFT process. The common steps for the all these implementations: the complex FFT, the inverse FFT overlapping techniques to avoid Gibbs phenomena Three types of frequency domain algorithm will be described: Hilbert transform method, Complex FFT method, and Spectral translocation method. 42

FT of Complex Functions If x(t) is a complex time function, i.e. x(t)=xr(t)+jxi(t) where xr(t) and xi(t) are respectively the real part and imaginary part of the complex function x(t), then the Fourier integral becomes + = + + + 2 j ft = + ( ) [ ( ) ( )] [ ( ) cos ( ) sin ] X f x t jx t e dt x t t x t t dt r i r i = + [ ( ) sin ( ) cos ] ( ) ( ) j x t t x t t dt R f jI f r i = j + + {cos } { ( ) ( )} F t 0 0 0 = + {sin } { ( ) ( )} F t 0 0 0 43

Properties of the Fourier transform for complex time functions Time domain (x(t)) Real Frequency domain (X(f)) Real part even, imaginary part odd Imaginary Real part odd, imaginary part even Real even, imaginary odd Real odd, imaginary even Real and even Real and odd Imaginary and even Imaginary and odd Complex and even Complex and odd Real Imaginary Real and even Imaginary and odd Imaginary and even Real and odd Complex and even Complex and odd 44

Interpretation of the complex Fourier transform If input of the complex Fourier transform is a complex quadrature time signal (specifically, a quadrature Doppler signal), it is possible to extract directional information by looking at its spectrum. Next, some results are obtained by calculating the complex Fourier transform for several combinations of the real and imaginary parts of the time signal (single frequency sine and cosine for simplicity). These results were confirmed by implementing simulations. 45

real part of complex FFT real part of complex FFT Case (1). x t X ( ) = = 1 1 = = = = t x t t } + + t t ( ) cos , ( ) sin {sin , } r i 0 0 = = 2 F jF {cos ( ) -1 -1 0 0 0 0 -Fs/2 0 -Fs/2 +Fs/2 +Fs/2 (5) (1) Case (2). x t X ( ) = = imaginary part of complex FFT imaginary part of complex FFT 1 1 = = = = t x t t } + + t t ( ) cos , ( ) sin , } r i 0 0 { sin = = 2 + + F jF {cos ( ) -1 -1 0 0 0 0 -Fs/2 0 -Fs/2 +Fs/2 +Fs/2 Case (3). x t X ( ) = = real part of complex FFT real part of complex FFT = = = = t x t t } + + t t ( ) cos { cos , ( ) sin {sin , } 1 1 r i 0 0 = = 2 + + F jF ( ) 0 0 0 Case (4). x t X ( ) = = -1 -1 0 -Fs/2 0 -Fs/2 +Fs/2 +Fs/2 (2) (6) = = = = t x t t } + + t t ( ) cos { cos , ( ) sin , } imaginary part of complex FFT imaginary part of complex FFT r i 0 0 1 1 { sin = = 2 F jF ( ) 0 0 0 Case (5). x t X ( ) = = -1 -1 0 -Fs/2 0 -Fs/2 +Fs/2 +Fs/2 = = = = t x t t } + + t t ( ) sin , ( ) cos {cos , } r i 0 0 real part of complex FFT real part of complex FFT = = 2 + + F jF j {sin ( ) 1 1 0 0 0 Case (6). x t X ( ) = = -1 -1 = = = = t x t t } + + t t ( ) sin , ( ) cos , } 0 -Fs/2 0 -Fs/2 +Fs/2 +Fs/2 (7) r i 0 0 (3) imaginary part of complex FFT imaginary part of complex FFT 1 1 { cos = = 2 F jF j {sin ( ) 0 0 0 Case (7). x t X ( ) = = = = = = t x t t } + + t t ( ) sin { sin , ( ) cos {cos , } -1 -1 r i 0 0 0 -Fs/2 0 -Fs/2 +Fs/2 +Fs/2 = = 2 F jF j ( ) 0 0 0 real part of complex FFT real part of complex FFT 1 1 Case (8). = = = = = = x t X t x t t } + + t t ( ) ( ) sin { sin , ( ) cos , } -1 -1 r i 0 0 0 -Fs/2 0 -Fs/2 +Fs/2 +Fs/2 (4) (8) imaginary part of complex FFT imaginary part of complex FFT { cos = = 2 + + F jF j ( ) 1 1 0 0 0 -1 -1 0 -Fs/2 0 -Fs/2 +Fs/2 +Fs/2 46

Frequency domain Hilbert transform algorithm DSP Algorithm Frequency domain complex HT algorithm D'(n) yf(n) SR H[S]R Q(n) -jS, w>0 +jS, w<0 CFFT IFFT H[S] yr(n) SI D(n) Q'(n) I 47

Complex FFT Method (CFFT) The complex FFT has been used to separate the directional signal information from quadrature signals so that the spectra of the directional signals can be estimated and displayed as sonograms. It can be shown that the phase information of the directional signals is well preserved and can be used to recover these signals. 48

DSP Algorithm SfR y (n) f D(n) SR IFFT S (w) f SfI CFFT Q(n) SrR SI y (n) r IFFT S (w) r SrI 49

= = + + = = = = + + + + + + + + s n ( ) D n ( ) jQ n s s n n H s n jH s [ j H s j s n { ( ) + + n s n r ( )} [ ( )]} ( ) { { ( ) ( ) [ ( )]} { [ ( )] f r f n jH s n ( )]} f f r r + + 2 ( ), 0 { ( ) ( )} { ( ) ( )}, 0 S S S j S S f f f r r = = = { ( )} ( ) F s n S {[ ( )} F s n + + 2 ( ), 0 j S { ( ) ( )} { ( ) ( )}, 0 S S j S S r f f r r ), 0 , 0 S 0 ( ), 0 S = ( ) S + = ( ) S ( 0 , 0 + + { ( )}, 0 { ( )}, 0 S S = = { ( )} { ( )} Sf Sf + + { ( )}, 0 { ( )}, 0 S S { ( )}, 0 { ( )}, 0 S S = = { ( )} { ( )} Sr Sr { ( )}, 0 { ( )}, 0 S S 50