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analysis. General Transform as problem-solving tool. s(t), S(f) : Transform Pair. time, t frequency, f F s(t) S(f) = F [s(t)]. synthesis. Frequency analysis: why?. Fast & efficient insight on signal’s building blocks. Simplifies original problem - ex.: solving Part. Diff. Eqns. (PDE).

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frequency analysis why

analysis

General Transform as problem-solving tool

s(t), S(f) : Transform Pair

time, tfrequency, f

F

s(t) S(f) = F[s(t)]

synthesis

Frequency analysis: why?
  • Fast & efficient insight on signal’s building blocks.
  • Simplifies original problem - ex.: solving Part. Diff. Eqns. (PDE).
  • Powerful & complementary to time domain analysis techniques.
  • Several transforms in DSPing: Fourier, Laplace, z, etc.
fourier analysis applications
Fourier analysis - applications
  • Applications wide ranging and ever present in modern life
    • Telecomms - GSM/cellular phones,
    • Electronics/IT - most DSP-based applications,
    • Entertainment - music, audio, multimedia,
    • Imaging, image processing,
    • Industry/research - X-ray spectrometry, chemical analysis (FT spectrometry), radar design,
    • Medical - EGG, heart malfunction diagnosis,
    • Speech analysis (voice activated “devices”, biometry, …).
fourier analysis tools

Periodic (period T)

FS

Discrete

Continuous

Aperiodic

FT

Continuous

**

Periodic (period T)

DFS

Discrete

Discrete

DTFT

Continuous

Aperiodic

**

DFT

Discrete

**

Calculated via FFT

Note: j =-1,  = 2/T, s[n]=s(tn), N = No. of samples

Fourier analysis - tools

Input Time Signal Frequency spectrum

fourier series fs

A periodic function s(t) satisfying Dirichlet’s conditions * can be expressed as a Fourier series, with harmonically related sine/cosine terms.

a0, ak, bk : Fourier coefficients.

k: harmonic number,

T: period,  = 2/T

For all t but discontinuities

(signal average over a period, i.e. DC term & zero-frequency component.)

* see next slide

Fourier Series (FS)

synthesis

analysis

Note: {cos(kωt), sin(kωt) }k form orthogonal base of function space.

fs synthesis
FS synthesis

Square wave reconstruction from spectral terms

Convergence may be slow (~1/k) - ideally need infinite terms.

Practically, series truncated when remainder below computer tolerance

( error). BUT … Gibbs’ Phenomenon.

gibbs phenomenon
Gibbs phenomenon

Overshoot exist @ each discontinuity

fourier integral fi

FourierIntegral Theorem

Any aperiodic signal s(t) can be expressed as a Fourier integral if s(t) piecewise smooth(1) in any finite interval (-L,L) and absolute integrable(2).

s(t) continuous, s’(t) monotonic

(1)

(3)

(2)

(3)

Complex form

Real-to-complex link

Fourier Transform (Pair) - FT

synthesis

analysis

Fourier Integral (FI)

Fourier analysis tools for aperiodic signals.

ft example

S(f) = 2 sMAX sync(2f)

Power Spectral Density (PSD) vs. frequency f plot. Note: Phases unimportant!

FT - example

FT of 2-wide square window

digital data formats

3

Integer part

Fractional part

Early computers (ex: ENIAC) mainly base-10 machines. Mostly turned binary in the ’50s.

a) less complex arithmetic h/w;

Benefits b) less storage space needed;

c) simpler error analysis.

Digital data formats

Positional number system with baseb:

[ .. a2 a1 a0.a-1 a-2 .. ]b = .. + a2 b2 + a1 b1 + a0 b0 + a-1 b-1 + a-2 b-2+ ..

Important bases: 10 (decimal), 2 (binary), 8 (octal), 16 (hexadecimal).

fixed point binary

3

Ex: 3-bit formats

15 14 ... 0

Unsigned integer

Offset-Binary

Sign-Magnitude

Two’s complement

7111

4111

3011

3 011

6110

3110

2010

2010

MSB LSB

5101

2101

1001

1001

4100

1100

0000

0000

Fractional point (DSPs)

3011

0011

0100

-1111

2010

-1010

-1101

-2110

1001

-2001

-2110

-3101

Sign bit

0000

-3000

-3111

-4100

Decimal equivalent

Binary representation

Fixed-point binary

Represent integer or fractional binary numbers.

NB: Constant gap

between numbers.

floating point binary 2

3

31 30 23 22 0

Precision

e

s

f

MSB LSB

Single (32 bits)

Double (64 bits)

Double-extended ( 80 bits)

e = exponent, offset binary, -126 < e < 127

s = sign, 0 = pos, 1 = neg

f = fractional part, sign-magnitude + hidden bit

Single precision range

Max = 3.4 · 1038

Min = 1.175 · 10-38

Coded number x = (-1)s · 2e · 1.f

NB: Variable gap between numbers.

Large numbers large gaps; small numbers small gaps.

Floating-point binary - 2

IEEE 754 standard

finite word length effects

3

Overflow : arises when arithmetic operation result has one too many bits to be represented in a certain format.

largest value

smallest value

Fixed point ~ 180 dB

Floating point ~1500 dB

Dynamic rangedB= 20 log10

High dynamic range wide data set representation with no overflow.

NB: Different applications have different needs.

Ex: telecomms: 50 dB; audio: 90 dB.

Finite word-length effects
dsp devices architectures
DSP Devices & Architectures
  • Selecting a DSP – several choices:
    • Fixed-point;
    • Floating point;
    • Application-specific devices(e.g. FFT processors, speech recognizers,etc.).
  • Main DSP Manufacturers:
    • Texas Instruments (http://www.ti.com)
    • Motorola (http://www.motorola.com)
    • Analog Devices (http://www.analog.com)
typical dsp operations

Pseudo C code

for (n=0; n<N; n++)

{

s=0;

for (i=0; i<L; i++)

{

s += a[i] * x[n-i];

}

y[n] = s;

}

Typical DSP Operations
  • Filtering
  • Energy of Signal
  • Frequency transforms
traditional dsp architecture
Traditional DSP Architecture

X RAM

Y RAM

a

x(n-i)

Multiply/Accumulate

Accumulator

y(n)

Most modern DSPs have more advanced features.

ti s dsp portfolio

‘C5000

(‘C54x)

‘C5x

‘C2000

(‘C20x, ‘C24x)

‘C1x ‘C2x

TI’s DSP Portfolio

‘C6000

(‘C62x, ‘C67x)

  • Power Efficient Performance
  • Wireless Telephones/IADs
  • Modems / Telephony
  • VoIP
  • .32ma/MIPS to sub 1V parts
  • $5 / 100 MIPS

‘C3x ‘C4x ‘C8x

  • Control Efficient
  • Storage
  • Brushless Motor Control
  • Flash Memory
  • A/D
  • PWM Generators
  • High Performance
  • Multi-Channel / Function
  • Comm Infrastructure
  • xDSL
  • Imaging, Video
  • VLIW architecture
  • 2400 MIPS +
  • Roadmap to 1 GHZ
slide17

PAST

PRESENT

C5000 TRANSFORMS QUALITY OF LIFE

Past

Present

Cellular Phones

Bulky

Sleek, Compact

Short Battery Life

Lasting Performance

Programmable, Multi-function

A Few, Fixed Functions

Internet Audio

Supports Multiple Standards, Field Upgradable

Supports One Standard

Long Design

Cycle

Fast Design Cycle

Digital Cameras

Personal and Portable Applications