Frequency analysis: why?

1. 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.

2. 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, …).

3. 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

4. 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.

5. 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.

6. Gibbs phenomenon Overshoot exist @ each discontinuity

7. 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.

8. 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

9. 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).

10. 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.

11. 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

12. 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

13. 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)

14. 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

15. Traditional DSP Architecture X RAM Y RAM a x(n-i) Multiply/Accumulate Accumulator y(n) Most modern DSPs have more advanced features.

16. ‘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

17. 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

19. C67x Architecture

23. DSP STARTER KIT (DSK)

24. C6711 DSK OVERVIEW