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Interpolation and Pulse Compression
09/27/2015 Brandon Ravenscroft EECS 800 Interpolation and Pulse Compression
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Interpolation (General)
Used when it is necessary to shift sample locations Interpolate between sample points of a function by βresamplingβ Implemented using convolution operation: π π₯ = π π π π β(π₯βπ) h(x) is interpolation kernel i = sample point π π₯ is continuous signal π π π is continuous signal sampled at i Kernel is typically even such that h(i-x) = h(x-i)
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Sinc Interpolation β π₯ =π πππ π₯ π π₯ = π π π π π πππ(π₯βπ)
Interpolation kernel is sinc function from Shannonβs sampling theorem Signal must be bandlimited (real) and satisfy Nyquist Interpolation kernel is : β π₯ =π πππ π₯ Interpolation operation is: π π₯ = π π π π π πππ(π₯βπ) Essentially a sinc reconstruction
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Sinc Interpolation β Frequency Domain
Spectrum of sampled signal repeats at intervals of the sampling frequency. Spectrum of sampled signal repeats at intervals of the sampling frequency. Only Baseband is needed, so apply an ILPF to sample spectrum. Rect in frequency domain ο³ sinc in time domain Multiplication in frequency domain ο³ convolution in time domain.
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Sinc Interpolation β Frequency Domain
Time shift property of the Fourier Transform πΉ π π‘β π‘ 0 = π π2ππ π‘ 0 πΊ(π) Multiply frequency domain signal by complex exponential In the Interpolation operation: π π₯ = π π π π π πππ(π₯βπ) π π π can be represented by a weighted impulse at i. Interpolation is a convolution operation Convolution in time is multiplication in frequency Impulse in time domain ο³ complex sinusoid in frequency domain Sinc in time domain ο³ rect function in frequency domain Conclusion: sinc interpolation is a frequency domain multiplication of a complex sinusoid and a rect function centered around i.
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Sinc Interpolation Example
fmax = 107 Hz fs = 214Hz
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Sinc Interpolation Example
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Sinc Interpolation Example
Missed Peaks!
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Sinc Interpolation Example - Reconstruction
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Sinc Interpolation β Intersample Interpolation
? g(9.3) Interpolation Kernel of Finite Size Centered at 9.3
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Sinc Interpolation β Intersample Interpolation
g(9.3)
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Sinc Interpolation β Intersample Interpolation
Infinite points in kernel needed to interpolate exactly. Sinc tapers off quickly, so finite window is all right ββ β π πππ π₯ ππ₯=1=ππππ‘(0) When truncating sinc kernel, normalization is needed since sum of weights no longer equals 1. βiβ is truncated window size πβ²(π₯)= π(π₯) π π πππ(π₯βπ) Normalization to keep signal power constant: πβ²(π₯)= π(π₯) π π πππ(π₯βπ) 2
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Sinc Interpolation β Procedural Notes
Sinc interpolation kernel can be weighted itself by a windowing function to reduce sharp transition ringing (Gibbβs phenomenon) Interpolation kernel can be pre-generated in subsample intervals and stored in a table. Accuracy of interpolation dependent upon kernel length Accuracy of sinc interpolation can be seen in frequency domain by comparing kernel spectrum to ILPF spectrum. Computational cost is proportional to kernel length. Interpolation of non-baseband and complex signals can be achieved by translating signal to baseband or translating baseband filter to signal center frequency. Multiply time domain signal by π Β±π2πβππ‘
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Uses Of Interpolation Range Doppler Algorithm(RDA)
Sinc Interpolation is used to perform range cell migration compensation (RCMC) to move targets into single range bin. F-k Migration (Omega-k migration) Stolt Interpolation used along the range frequency axis to correct for range curvature. Time Domain Correlation (TDC) Applying a matched filter to a signal input. Calibration
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Pulse Compression Time domain compression of a pulse signal via frequency modulation. Increases SNR and Range Resolution. SNR Increased by a factor of the time-bandwidth product: π΅π π π = ππ 2 = π 2π΅ Tx pulse is frequency/phase modulated and Rx signal is correlated with this. Matched filtering is frequently used in detecting incoming signals. Matched filter is complex conjugate of received signal.
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Pulse Compression β Matched Filtering
Consider a received pulse signal of duration T: π π (π‘)=ππππ‘ π‘ π π πππ π‘ 2 A matched filter of duration T: β(π‘)=ππππ‘ π‘ π π βπππ π‘ 2 The matched filter output is: π ππ’π‘ π‘ = π π (π‘)ββ(π‘)= πβ π‘ ππππ‘ π‘ 2π π πππ[(ππ‘ πβ π‘ ] Slowly Varying Envelope: πβ π‘ ππππ‘ π‘ 2π Rapidly Varying Portion: π πππ[(ππ‘ πβ π‘ ]
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Slowly Varying Envelope: πβ π‘ ππππ‘ π‘ 2π
π΅=10 ππ»π§ π=10 Β΅π π΅π=100
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Rapidly Varying Portion: π πππ[(ππ‘(πβ|π‘|)]
π΅=10 ππ»π§ π=10 Β΅π π΅π=100
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Composite Matched Filter Output: π ππ’π‘ (π‘)
π΅=10 ππ»π§ π=10 Β΅π π΅π=100
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Pulse Compression Comparison of Bπ
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Pulse Compression - Notes
When BΟ β₯ 100, Matched Filter Output Simplifies to: π ππ’π‘ (π‘)=π πππ(πππ‘) Signal and matched filter can have different lengths, math works out to similar result with extra quadratic phase term: π πππ π‘ 2 which is usually negligible
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Questions/Discussion?
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