Name: zero-inserted signal reconstruction
Uploaded: 2024-09-26T13:17:07.000Z
Duration: 1 min 22 s
Channel: JoVE

K12 Standards

Next Generation Science Standards

High School

Physical Science

waves-and-their-applications-in-technologies-for-information-transfer

Waves and Their Applications in Technologies for Information Transfer

HS-PS4-2

Upsampling

Weźmy pod uwagę zdziesiątkowany sygnał o zmniejszonym zakresie częstotliwości ze względu na niższą częstotliwość próbkowania.
Wstaw zera między każdą próbką, aby zwiększyć próbkę, co spowoduje wprowadzenie powtarzających się replik spektralnych w odstępach czasu określonych przez nową częstotliwość Nyquista.
Przepuść sekwencję z wprowadzonym zerem przez filtr dolnoprzepustowy z częstotliwością odcięcia na nowym limicie Nyquista. Filtr ten tłumi repliki o wyższej częstotliwości, zachowując tylko oryginalne składowe częstotliwości.
Przefiltrowane wyjście wytwarza sygnał o wyższej częstotliwości próbkowania, skutecznie utrzymując oryginalny sygnał i odwracając proces próbkowania w dół.
Weź sekwencję z transformatą Fouriera pokazującą niezerowe wartości od -4π/9 do +4π/9.
Zmniejsz próbkowanie o dwa, co daje widmo od -8π/9 do + 8π/9.
Upsampamplizuj go o osiem, kompresując transformatę Fouriera tak, aby zakres od -π/9 do + π/9.
Zmniejsz próbkowanie o dziewięć, skalując transformatę Fouriera do zakresu od -π do +π.
Połączenie upsamplingu przez cztery i downsamplingu przez dziewięć daje maksymalne próbkowanie w dół bez aliasingu.

upsampling

Zarządzanie częstotliwościami próbkowania sygnału jest niezbędne aby zachować integralność sygnału podczas przetwarzania. Sygnał poddany decymacji, charakteryzujący się mniejszym zakresem częstotliwości ze względu na niższą częstotliwość próbkowania, można poddać upsamplingowi, wstawiając zera między każdą próbką. Ten proces upsamplingu rozszerza oryginalne widmo i wprowadza powtarzane repliki widmowe w odstępach dyktowanych przez nową częstotliwość Nyquista. Aby udoskonalić tę sekwencję z wstawionymi zerami, przepuszcza się ją przez filtr dolnoprzepustowy z częstotliwością odcięcia ustawioną na nowym limicie Nyquista. Ten filtr tłumi repliki o wyższej częstotliwości, zachowując tylko oryginalne składowe częstotliwości.


Wynikiem tego procesu filtrowania jest sygnał o wyższej częstotliwości próbkowania, który skutecznie odwraca procedurę decymacji. Na przykład rozważ sekwencję z transformacją Fouriera wykazującą wartości niezerowe od −2π/9 do 2π/9. Jeśli ta sekwencja zostanie zmniejszona o współczynnik czterech, jej widmo rozciąga się od −8π/9 do 8π/9. Następnie, podwyższenie częstotliwości próbkowania sekwencji o współczynnik dwóch kompresuje transformację Fouriera, która teraz mieści się w zakresie od −π/9 do π/9.


Dalsze podwyższenie częstotliwości próbkowania tej podwyższonej częstotliwości próbkowania o dziewięć skaluje transformację Fouriera, aby rozszerzyć ją od −2π/9 do 2π/9. Ta kombinacja podwyższenia częstotliwości próbkowania o dwa i podwyższenia częstotliwości próbkowania o dziewięć jest równoważna podwyższeniu częstotliwości próbkowania o współczynnik 9/2, co pozwala na osiągnięcie maksymalnego obniżenia częstotliwości próbkowania bez wprowadzania aliasingu.


Proces podwyższenia częstotliwości próbkowania poprzez wstawianie zer i późniejsze filtrowanie dolnoprzepustowe, a następnie precyzyjne kombinacje podwyższenia częstotliwości próbkowania i podwyższenia częstotliwości próbkowania, umożliwiają efektywne zarządzanie częstotliwościami próbkowania sygnału. Ta metoda zapewnia zachowanie integralności oryginalnego sygnału, zapobiegając aliasingowi i zniekształceniom przy jednoczesnym dostosowaniu do różnych wymagań próbkowania.


Takie techniki są ważne w przetwarzaniu sygnałów cyfrowych, gdzie równowaga między wydajnością próbkowania a wiernością sygnału jest najważniejsza. Poprzez ostrożne dostosowywanie częstotliwości próbkowania za pomocą tych procesów możliwe jest zachowanie istotnych cech oryginalnego sygnału, ułatwiając dokładne przetwarzanie sygnału i rekonstrukcję w różnych dziedzinach technologii, w tym komunikacji, inżynierii dźwięku i kompresji danych.

Weźmy pod uwagę zdziesiątkowany sygnał o zmniejszonym zakresie częstotliwości ze względu na niższą częstotliwość próbkowania.
Wstaw zera między każdą próbką, aby zwiększyć próbkę, co wprowadza ...

Education

JoVE Core

Electrical Engineering

Elektrotechnika

Unit 1

Jednostka 1

Sampling

Próbkowanie

KEY TERMS AND CONCEPTS

KLUCZOWE TERMINY I POJĘCIA

zero-inserted signal reconstruction

Zero-inserted signal reconstruction is a digital signal processing method used to raise a signal’s sampling rate while keeping its important frequency content. A decimated signal has fewer samples and a narrower frequency range. To upsample it, zeros are inserted between the original samples.

This zero-stuffed sequence changes the spectrum. It creates repeated spectral replicas at intervals set by the new Nyquist frequency, which is the highest frequency allowed by the new sampling rate. A lowpass filter is then applied with a cutoff at that new Nyquist limit. The filter reduces the higher-frequency replicas and preserves the original signal components.

The filtered result is a higher-rate signal that effectively reverses the earlier downsampling step. The transcript also shows how sampling-rate changes can be combined. A signal with non-zero Fourier values from −2π/9 to 2π/9 becomes wider after downsampling by four. Upsampling that signal by two compresses the Fourier transform to −π/9 to π/9, and a later downsampling by nine extends it back to −2π/9 to 2π/9.

That sequence of upsampling by two and downsampling by nine is equivalent to downsampling by 9/2. It gives the maximum downsampling possible without aliasing, which is frequency overlap that can distort a signal. These steps are important in communications, audio engineering, and data compression, where signal fidelity must be protected while sampling rates are adjusted.

Zero-Inserted Signal Reconstruction

sampling theorem

Sampling Signals for Digital Use

Sampling signals for digital use turns a continuous-time signal, x(t), into a discrete-time signal. This step is essential when a real-world signal must be stored or processed by a digital system. A train of impulses sets the sampling interval, and the sampling frequency describes how often the signal is measured. The sampling process works by multiplying the continuous-time signal by the impulse train. This creates a series of discrete impulses called the sampled signal. Engineers then study...

sampling continuous time signal

Impulse-Train Sampling and Zero-Order Hold

Impulse-train sampling and zero-order hold are two key steps in digital signal processing. Together, they show how a continuous-time signal can be turned into samples and then built back into a continuous-looking waveform. Impulse-train sampling uses a periodic impulse train. This train is a series of delta functions placed at regular intervals set by the sampling period. When the continuous-time signal is multiplied by the impulse train, the result is a set of impulses whose amplitudes match...

reconstruction of signal using interpolation

reconstruction-of-signal-using-interpolation

Signal Reconstruction with Low-Pass Filtering

Signal reconstruction with low-pass filtering helps turn a sampled signal back into a smooth continuous signal. When a continuous signal is sampled with a period T, the sampled version creates replicas of the original spectrum in the frequency domain. These replicas are spaced by the sampling frequency. A zero-order hold can be used to process the sampled signal. This method keeps each sample value until the next sampling period, so the output becomes piecewise constant. In the frequency...

aliasing

Sampling Signals Without Aliasing

Sampling a signal at the right rate is essential for accurate signal reconstruction. A time-domain signal can be analyzed with its Fourier transform, which reveals its spectrum. When the signal is sampled, the spectrum becomes multiple scaled replicas in the frequency domain. The spacing between these replicas depends on the sampling frequency. If the sampling frequency is below the Nyquist rate, the replicas overlap. The Nyquist rate is twice the highest frequency in the original signal. Once...

downsampling

Decimation and Sampling Rate Reduction

Decimation reduces a sampled sequence by keeping every N-th sample. At those integer multiples of N, the original and reduced sequences match. This makes the sequence shorter and more efficient for digital signal processing. The Fourier transform of a decimated sequence is not just a simple copy of the original. It becomes a scaled and shifted combination of the original spectrum. In effect, the reduced sequence reflects the same frequency content, but in a form shaped by the new sampling...

Zero-inserted signal reconstruction is a digital signal processing method used to raise a signal’s sampling rate while keeping its important frequency content. A decimated signal has fewer samples and a narrower frequency range. To upsample it, zeros are inserted between the original samples. This zero-stuffed sequence changes the spectrum. It creates repeated spectral replicas at intervals set by the new Nyquist frequency, which is the highest frequency allowed by the new sampling rate. A...

bandpass sampling

Sampling Narrow Frequency Bands

Bandpass sampling is a signal processing method for signals whose energy is concentrated in a narrow frequency band. Such a signal is called a bandpass signal. The method works when the sampling rate is greater than twice the signal bandwidth, which helps prevent aliasing. A bandpass signal has a spectrum with a lower limit, ω1, and an upper limit, ω2. Its spectrum is nonzero only between those two frequencies. To sample it, the time-domain signal is multiplied by an impulse train. This...

key terms and concepts

Learn how zero insertion and lowpass filtering rebuild a signal at a higher sampling rate while reducing spectral replicas and avoiding aliasing.

Zero-Inserted Signal Reconstruction | JoVE K12

JoVE Core Electrical Engineering offers easy-to-understand animated videos that cover fundamental concepts of electric circuits along with circuit analysis techniques. It incorporates first-order and second-order circuits using operational amplifiers and circuit transformation concepts. The inclusion of practical design examples and real-world applications through scientist-in-action videos adds significant educational value, making complex concepts accessible.

electrical engineering

HS-PS4-5

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HS-PS4-1

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HS-PS4-3

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HS-PS4-4

Zero-Inserted Signal Reconstruction

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