# Signal Sampling and Reconstruction • Theory • Antialiasing and

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```					ESE 347, Digital Signal Processing: Implementation

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Signal Sampling and Reconstruction • Theory • Antialiasing and Smoothing Filters • Signal Conversion

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Sampling and Reconstruction Theory Two Steps: • Multiply by train of δ-functions • Pick off numbers representing samples

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Original Signal: g(t) Sampled Signal: gS (t) = Sequence of Samples: (gn ) = (g(nT )) where T is the sampling period and fS = 1/T is the sampling frequency.
∞

n=−∞

∑

g(nT )δ(t − nT )

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Effect of sampling in frequency domain: GS (2π j f ) = 1/T

k=−∞

∑

∞

G(2π j( f − k fS ))

fS /2 is sometimes called the folding frequency, and ωS = 2π fS is called the sampling angular frequency. WARNING!1! Nonlinear operations internal to the signal processing can increase the bandwidth, and require much higher sampling rates. WARNING!2! Aliased components are not just a mathematical abstraction, but very real; you can see them on an oscilloscope, and hear them. WARNING!3! Higher sampling rates can also be needed to reduce phase-shift, for example, in a feedback loop.

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Classical Sampling Criterion: Must Sample at a rate higher than the Nyquist rate = 2 fMAX

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To reconstruct: use lowpass ﬁlter Reconstruction is also possible in other cases: • Bandpass • Periodic

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In time-domain: gR (t) =

n=−∞

∑

∞

g(nT )l(t − nT )

where l(t) is the impulse response of the reconstruction (smoothing) ﬁlter.

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Antialiasing (Input) and Smoothing (Output) Filters: • Needed from sampling theorem • To reject spurious signals and noise at input • To reject aliased components at output Problem: Need high order analog ﬁlter, especially at output

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Output: • Upconvert, ﬁlter digitally, high-rate D/A converter, • Upconvert, high-rate D/A converter, switched-capacitor ﬁlter, ﬁnal simple smoothing ﬁlter.

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Input: • Gentle analog ﬁlter, very high sampling rate – low accuracy • Use high-frequency, low word-width Digital lowpass ﬁlter • Use DSP to Down-convert to low sampling rate, high word-width – noise-shaping

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Track and Hold Ampliﬁer: • Tracks signal until strobed, and holds constant value until released. • Stores charge on capacitor • Properties – Aperture – Acquisition time – Step – Droop – Jitter

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A/D Converters: • Successive Approximation • Flash • Sigma-Delta • Dual-Slope, Counting, usually too slow

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D/A Converters: Usually switched current source, with zero-order hold. Amplitude Response: sin(x)/x

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