Algorithms, Applications and Systems for Digital

Signal Processing

Patrick Gaydecki

School of Electrical and Electronic EngineeringThe University of Manchester

PO Box 88Manchester M60 1QD

United Kingdom

www.sisp.manchester.ac.uk/dsp.shtml

patrick.gaydecki@manchester.ac.uk

[UK-44] (0) 161 306 4906

Signal Processing Seminar

Developing Good Practice in Metrology Applications

30th November 2005 - NPL, Teddington, UK

Overview of Presentation

• The advantages of real-time DSP in weak signal detection

• Typical real-time DSP hardware and software specifications

• Finite impulse response (FIR) design and applications

• Infinite Impulse response design and applications

• Inverse filter design and applications

• Case study: weak AC magnetic field detection

Characteristics of Real-Time DSP Systems

• DSP offers flexibility, allowing a single platform to be rapidlyreconfigured for different applications

• Operations such as modulation, phase shifting, signal mixing and delaying are simply performed in software

• System performance is far more accurate than equivalent analogue systems

• However, considerable intellectual investment is required to design and program DSP platforms

DSP Hardware and Software Specifications

(Signal Wizard, Developed at UoM)

• 24-bit codec resolution (1 part in 16777216, or 144 dB)

• Variable sample rate extending to 196 kHz

• Processor power of 100 million multiplication-accumulations per second (MMACS)

• Dual channel operation for signal referencing and mixing

• Non-volatile memory for retention of settings

• Standard PC interface

• Easy FIR, and IIR filter design, with other processing functions including mixing and phasing

An

ato

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f a T

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ica

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evic

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OP

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de

Me

mo

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em

ory

Da

taD

ata

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mo

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em

ory

synchronous serialsynchronous serial

interfaceinterface

serial serial

communicationscommunications

interfaceinterface

parallel parallel

communicationscommunications

interfaceinterface

bu

s s

tructu

reb

us s

tructu

re

Components of the Real-Time DSP System

DSPDSP

FlashFlash

memorymemory

24

24-- b

it d

ua

lb

it d

ua

l

ch

an

ne

l co

dec

ch

an

ne

l co

dec

Inte

rface

Inte

rface

Vin

Vout

CodecCodec SSI of DSPSSI of DSP

Frame clock

Bit clock

Data

TS1TS1 TS2TS2 TS3TS3 TS4TS4

DataData

Frame clock

Bit clock

Data in normal mode

Data in network mode

The Inherent Versatility of the Synchronous

Serial Interface

550855001000

1100195250500

2750450100200

550088850100

1375024892040

2750048881020

550009888510

Filter coefficients, DSP56321

Filter coefficients, DSP56309

Nyquist frequency,

kHz

Sample frequency,

kHz

FIR Performance Figures for

DSP56309 and DSP56321

Final Realized Hardware

(Signal Wizard)

FIR and IIRFIR and IIR

design areadesign area

Graphical Graphical

display of filterdisplay of filter

Hardware control: download, gain, Hardware control: download, gain,

adaptive, delay, mixing etc.adaptive, delay, mixing etc.

Final Realized Software (Signal Wizard)

not in practiceyesArbitrary response

lowhighComputational load

Output signal

Transfer function

noyesUnconditional stability

yes

easy

good

yesAnalogue equivalent

unwieldyDesign ease

poorWord length immunity

∑−

=

−=1

0

][][][M

k

knxkhny

H zY z

X zh n z

n

n( )( )

( )[ ]= =

=

∞

−∑0

∑∑==

−−−=M

k

N

k

knykbknxkany10

][][][][][

∑

∑

=

−

=

−

−−

−−

+

=+++

+++=

N

n

n

M

m

m

n

m

znb

zma

znb...zb

zma...zaazH

1

0

1

1

][1

][

][]1[1

][]1[]0[)(

FIRProperty / Filter type IIR

Basic Linear Filter Theory: τττ dtxhty )()()( −= ∫∞

∞−

Finite Impulse Response (FIR) Filter Design

using the Frequency Sampling Approach

∑−

=

−=1

0

][][][M

k

knxkhny

[ ]

=

=

<<=

<<=

= −

elsewhere0][

elsewhere0][

][][

][][

][][][1

kH

kH

fkfkbkH

fkfkakH

kHFnwnh

i

r

li

lr

Arbitrary filter impulse response h[n] obtained by:

Output signal y[n] obtained by discrete-time convolution:

Windowing: Blackman v Rectangular

Impulse responseImpulse response

withwith

Blackman windowBlackman window

Frequency response Frequency response

withwith

Blackman windowBlackman window

Impulse responseImpulse response

withwith

rectangular windowrectangular window

Frequency response Frequency response

withwith

rectangular windowrectangular window

Precise phase control for each

harmonic, to a resolution of

0.0001 degree. Application:

real-time Hilbert transform.

{ } { })()()( 1 ωhh YFtyHty−==

<

>−=

0),(

0),()(

ωω

ωωω

jY

jYYh

Phase Control

Infinite Impulse Response (IIR) Filter Design

∑∑==

−−−=M

k

N

k

knykbknxkany10

][][][][][

Digital Emulation of Analogue Networks I:

Laplace to z-domain mapping

sL

1/sC

R

vin(t)

vout(t)

jω

-jω

σ-σ

j

-j

1-1

1)(

2 ++=

sRCLCs

sRCsH

s-domain

z-domain

ωσ js +=

TjTeez

ωσ=

1)(

2 ++=

sRCLCs

sRCsH

+

−=

−

−

1

1

1

12

z

z

Ts

11

12

1

12

1

12

)(

1

12

1

12

1

1

+

+

−+

+

−

+

−

=

−

−

−

−

−

−

z

zRC

Tz

zLC

T

z

zRC

TzH

LCT

b

T

RCa

22

2

=

=

11

1

1

1

1

1

1

1

)(

1

1

1

1

1

1

1

1

+

+

−+

+

−

+

−

+

−

=

−

−

−

−

−

−

−

−

z

za

z

z

z

zb

z

za

zH

21

2

)1()22()1()(

−−

−

+−+−+++

−=

zabzbba

azazH

)1(

)1(

)1(

)22(

)1(

2

1

0

++

+−=

++

−=

++=

ba

ab

ba

b

ba

a

α

α

α

2

2

1

1

2

0

1

)1()(

−−

−

++

−=

zz

zzH

αα

α

]2[]1[]2[][][ 2100 −−−−−−= nynynxnxny αααα

Substitutions:

Digital Emulation of Analogue Networks II:

The Bilinear z-transform (BZT)

Final difference equation:

Digital Emulation of Analogue Networks III:

Real-Time Performance

-15

-10

-5

0

5

10

15

0 0.005 0.01 0.015 0.02

Time, s

Vo

lts

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0 2000 4000 6000 8000 10000

Frequency, Hz

Rela

tive m

ag

nit

ud

e

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0 0.005 0.01 0.015 0.02

Time, s

Vo

lts

0.00

0.01

0.01

0.02

0.02

0.03

0.03

0.04

0 2000 4000 6000 8000 10000

Frequency, Hz

Rela

tive m

ag

nit

ud

e

Design

Realised

IIR Comb Filters and Pole-Zero Placement

))((

))((

))((

))((

))((

))(()(

12

12

32

32

10

101

0 −−

−−−

= −−

−−×

−−

−−×

−−

−−=

−

−= ∏

nn

nnn

k k

k

pzpz

zzzz

pzpz

zzzz

pzpz

zzzz

pz

zzzH KK

))((

))(()(

10

10

pzpz

zzzzzH

−−

−−=

111

110

001

000

βα

βα

βα

βα

jp

jp

jz

jz

−=

+=

−=

+=

2

1

2

11

2

0

2

00

βαε

βαε

+=

+=

[ ][ ][ ][ ] 2

11

2

0

1

0

1111

0000

21

21

)()(

)()()(

−

−−

+−

+−=

−−+−

−−+−=

z

zz

jzjz

jzjzzH

εα

εα

βαβα

βαβα

]2[]1[2]2[]1[2][][ 1100 −−−+−+−−= nynynxnxnxny εαεα

0.00E+00

5.00E+00

1.00E+01

1.50E+01

2.00E+01

2.50E+01

3.00E+01

0 2000 4000 6000 8000

Frequency, Hz

Rela

tive m

ag

nit

ud

e

0.00E+00

5.00E+00

1.00E+01

1.50E+01

2.00E+01

2.50E+01

3.00E+01

0 2000 4000 6000 8000

Frequency, Hz

Rela

tive m

ag

nit

ud

e

-600

-400

-200

0

200

400

600

0 0.01 0.02 0.03 0.04 0.05

Time, s

Vo

lts

-600

-400

-200

0

200

400

600

0 0.01 0.02 0.03 0.04 0.05

Time, s

Vo

lts

j

-j

1-1

Inverse Filters: Design and Applications

∑−

=

− −=1

0

1 ][][~

][M

k

knxkhny

Signal Shape Reconstruction

Essential Equations Describing Time Domain Deconvolution (Inverse Filtering)

for Real-Time and Off-Line Processing

)()()()(*)()( ωωω HXYthtxty =⇔=

= −

)(

)()( 1

ω

ω

H

YFtx

)()(*)()( tsthtxty +=

−= −

)(

)(

)(

)()( 1

ω

ω

ω

ω

H

S

H

YFtx

<<

<=

otherwises

Hif

s

,10

)(

1,0 γ

ω

+= −−

sHFth

)(

1)(

~ 11

ω

)(~

*)()( 1thtytx

−=

Simple Fourier domain scheme (rarely successful):

Fourier domain scheme with noise estimate:

Time domain scheme with noise estimate (surprisingly useful):

Finally:

Signal Shape Reconstruction in Practice

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

-0.003 -0.002 -0.001 0 0.001 0.002 0.003 0.004

time (s)

volts

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

-0.003 -0.002 -0.001 0 0.001 0.002 0.003 0.004

time (s)

volts

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

-0.003 -0.002 -0.001 0 0.001 0.002 0.003 0.004

time (s)

volts

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

-0.003 -0.002 -0.001 0 0.001 0.002 0.003 0.004

time (s)

volts

Signal comprising three impulses

Signal after inverse filtering in real-timeSignal after low-pass distortion

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

-0.003 -0.002 -0.001 0 0.001 0.002 0.003 0.004

time (s)

volts

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

-0.003 -0.002 -0.001 0 0.001 0.002 0.003 0.004

time (s)

volts

Case study 1: Detection of AC magnetic fields

propagated through a ferrous steel boundary:

The Skin Effect

)(. δ

ωδ

dtj

d

s eeJJ−−

=

σµωµδ

r0

2=

The strength of both the eddy currents and the associated magnetic field fall rapidly with depth in ferrous

materials. The equation which describes the fall in current density is given by:

Where Js is the current density on the surface, d is the depth within the material, δ is the skin depth, ω is

the angular frequency and J is the is the current density at depth d. The skin depth for a given material is

governed by the relationship:

where σ is the conductivity of the conductor or target, µr its relative permeability and µ0 is the absolute

permeability of a vacuum.

digital functiondigital function

generatorgenerator

InstrumentationInstrumentation

amplifieramplifier

(x 400)(x 400)

DigitalDigital

OscilloscopeOscilloscope

powerpower

amplifieramplifier

super super

narrowbandnarrowband

filterfilter

digital gaindigital gain

(x 4096)(x 4096)

DSPDSP

8.99 × 10-122.20 × 10-80.1144.09 × 10-413.0

7.89 × 10-114.57 × 10-70.1371.73 × 10-49.0

1.03 × 10-83.33 × 10-50.1943.1 × 10-44.5

Receivedfluxdensity, T

AttenuationSkin depth, mm

Transmittedfluxdensity, T

Frequency, kHz

Steel properties:

Relative permeability: 250

Conductivity: 6.0 × 106 Sm-1

Mild steel plate

Mild steel enclosure

Experimental Configuration

Laboratory System

-100

-90

-80

-70

-60

-50

-40

-30

-20

-10

0

0 5000 10000 15000 20000 25000

Frequency, Hz

Am

plit

ude,

dB

-0.923870.382680.923870.382689.0

-0.555560.831460.555560.831464.5

p1, imaginaryp

1, realp

0, imaginaryp

0, realCentre Frequency, kHz

Pole locations for two IIR filters

IIR Filter Frequency Response at 9 kHz

Line Scan Results at 4.5 kHz and 9 kHz

0

100

200

300

400

500

600

700

800

900

0 200 400 600 800 1000

Distance across plate, mm

mV

(R

MS

)

1.00E-07

1.00E-06

1.00E-05

1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+00

0 5000 10000 15000 20000 25000

Frequency, Hz

Log a

mplit

ude

0

200

400

600

800

1000

1200

1400

0 200 400 600 800 1000

Distance across plate, mm

mV

(R

MS

)

1.00E-07

1.00E-06

1.00E-05

1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+00

0 5000 10000 15000 20000 25000

Frequency, Hz

Log a

mplit

ude

4.5 kHz 9 kHz

Case study 2: Detection of low amplitude ultrasonic

pulses propagated through seawater

via a steel structure

7 m structure being lowered into the dockat Liverpool, UK

Location of transmitter

InstrumentationInstrumentation

amplifieramplifier

(x 400)(x 400)

DigitalDigital

OscilloscopeOscilloscope

super super

narrowbandnarrowband

filterfilter

digital gaindigital gain

(x 2048)(x 2048)

DSPDSP

MicrocontrolledMicrocontrolled

pulserpulser

TransmittingTransmitting

transducertransducer

ReceivingReceiving

transducertransducer

100 m100 m

650 nVPeak received signal

Geometric (1/r2)Attenuation

hemisphericalDivergence angle

20 VTransmission amplitude

40 kHzFrequency

Tone burstSignal type

Experimental Configuration

Time

Am

plit

ude

Time

Am

plit

ude

Time

Am

plit

ude

Typical Results

(a) Detail of original received signal

degraded by noise.

(b) Detail of received signal,

recovered by super narrowband

filter.

(c) Complete tone burst signal

detected after transmission

through water, recovered using

a super narrowband IIR filter.

(a)

(b)

(c)

Conclusions

• Flexibility ensures that real time DSP is suitable for many applications in weak field detection

• Software-realized processing yields very significant improvements with regard to stability, precision and repeatability

• Multiple frequencies can be interrogated simultaneously, unlike lock-in detection

• At least a factor of ten cheaper than lock-in detection

• New generation DSP devices are extending the range of signal frequencies over which real time discrete processing can be applied

• However: DSP system design requires investments in time and effort

Acknowledgements

With thanks to Bosco Fernandes, Rito Mijarez, Graham Miller and Muhammad Zaid. We also acknowledge the EPSRC for financially

supporting this work.