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Software Defined Radio Lec 7 – Digital Generation of Signals Sajjad Hussain, MCS-NUST.

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Outline for Today’s Lecture Digital Generation of Signals 1. Introduction 2. Comparison to Analog generation 3. DDS Techniques 4. Analysis of Spurious Contents 5. Band-pass Signal Generation 6. Performance of DDS Systems 7. Generation of Random Numbers 8. ROM compression techniques

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Generation of Random Sequences Random sequences are needed in a variety of communication applications scrambling, bit- synchronization, spreading, security etc. Spreading Use of different codes for same freq. Scrambling Help maintain synchronization and adding randomness.. Ideal binary random sequence (infinite length, identically distributed RV ) vs. PN sequences (finite length)

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Type of Sequences Most common technique for generating PN sequences use of binary digital linear feedback shift register Maximum Length Sequences Sequences with a maximum-period are called max length seq. m- sequences Shift register with 2 m -1 period -> polynomial should be primitive-> irreducible-> cannot be factored into product of polynomials with binary coefficients and degrees of at-least 1 If N = 2 m -1 is the period of sequence y(n), then the periodic auto-correlation function is

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Gold Sequences Composite codes with good and well-defined cross-correlation properties Generated by using ‘preferred m-sequences’ m-sequences with certain specific correlation properties Modulo-2 sum of 2 preferred m-sequences Same length as that of input codes A different code is generated by shifting one of the codes Thus construction of 2 m -1 codes from pairs of m-stage shift registers Though constructed from m-sequences, are not maximal sequences Codes can be selected with bounded cross-correlation properties

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Gold Code Generator

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Gold Codes with bounded auto- correlation

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Randomization with Wheatley procedure Used for removal of harmonic spurs If removal not possible, spreading energy in all harmonics is useful – Wheatley procedure high noise floor with few strong harmonics Randomly varying (dithering) the periods of output, while keeping the average of these periods unchanged The method consists of adding a sequence of random numbers to the contents of the accumulator in a prescribed manner to convert harmonic signals into a continuous noise floor, whose level is much lower than that of harmonic signals At each clock-cycle a RV is added – 0:Δ r -1 Introduces un-correlated phase noise

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Wheatley Procedure

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Effect on Spectrum because of Wheatley Procedure

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ROM Compression Spurious signals are one of the main drawbacks of DDS system, especially those caused by phase- truncation – spurious harmonic signals phase-truncation – to avoid a very large ROM Phase-truncation can be avoided if it was possible to compress more information into the ROM One simple compression approach takes advantage of the symmetry of sine-wave store only one quadrant of information eliminates 75% of the normal memory requirements Other techniques along-with the sine-symmetry – interpolation-based

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Interpolation using Taylor Series Expansion Certain values of sine function are stored in ROM and the values in-between these angles can be interpolated using Taylor series expansion

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Interpolation using two terms of power series

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Effect of using four-terms of power series

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Effect of using seven-terms of power series

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Interpolation using trignometric identities Using trigonometric identities to find the values between the exact known values Most of these methods work only well when the deviation from the known angle is very small Hutchison Algorithm : Division of values of sine function in first quadrant into ‘coarse’ and ‘fine’ ROM Trig. Identities can then be used to generate the sine values for any angle θ by decomposing it to values contained in the coarse and fine ROM No. of bits addressing the ROM are divided into C coarse bits (for θ C ) and F fine bits (for θ F ) If θ= θ C + θ F

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Example : ROM size savings using Hutchison algorithm For an accumulator (address) width = 12 bits and ROM width = na = 12 bits total no. of bits stored is 2 12 * 12 = 49,152 Same resolution can be obtained using a lesser no. of stored bits by Hutchison algorithm If C= 8 bits and F = 4 bits Total no. of bits required for storing 2 4 * 12 + 2 8 * 12 = 3,264 bits

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Sunderland algorithm An improvement over Hutchison algorithm and divides the phase-angle into three parts, thus using 3 ROMs θ= θ C + θ s + θ F The coarse angles are defined in the first quadrant of a sine-wave from 0 to π/2, divided into 2 C equal angles. The Sunderland angle is defined as one of coarse angles divided into 2 S equal angles. Finally, the fine angle is defined as one of the Sunderland angles divided into 2 F equal angles

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Sine-Phase Difference Algorithm Approach Introduces a way to reduce the storage requirements for the quarter-wave sine function. The idea is to store f(θ) = sin (πθ/2) – θ, instead of sin (πθ/2) The variation in the function f(θ) values is small, and thus a small LUT (as many as two bits saving for storing amplitude values) can be used to represent f(θ) and sin (πθ/2) can be easily calculated from f(θ) Sine LUT propagation delay is also reduced, increasing the maximum clock freq. of DDS

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Modified Sine-Phase Difference Algorithm Approach – Parabolic Approximations In this approach, a parabola is used to approximate the sinusoid of the sine half-period To generate the same sine wave, the sine parabola difference approximation uses a more narrow range of values (saves as many as 4 bits of memory word-length) than the sine-phase difference approach Additional hardware to generate corresponding parabola values at ROM output can be easily implemented without significant complexity

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Example : Qualcomm’s Q2240 Direct Digital Synthesizer Suited for needs of wireless comm. And complex waveform synthesis Max freq. 100 MHz (5V) or 60 MHz (3.3 V) 31-bit Freq. Control Register (FCR), 32-bit phase-accumulator, 14– bit address output and 12-bit sine LUT 14-bit phase output resolution 12-bit output resolution The latched FCR value is accumulated in the phase-accumulator in every clock-cycle The LUT can be by-passed, ending the 14 MSBs of the phase- accumulator directly to the output The unused sine LUT is de-activated to reduce power consumption

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Block Diagram of Qualcomm’s DDS – Q2240I-3S1

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Conclusion DDS in comparison to analog approaches provide Flexibility Fine freq. resolution Fast response time Ease-of-manufacturing and testing Robustness to environmental changes Most DDS ACC + ROM + DAC Issue in DDS Design Spurious signal removal Hybrid designs ROM-size constraints compression techs., trig. Identities. Etc.

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