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

サンプリング レートが低いために周波数範囲が縮小されたデシメーション信号について考えてみましょう。
アップサンプリングする各サンプル間にゼロを挿入すると、新しいナイキスト周波数によって決定された間隔でスペクトル レプリカが繰り返されます。
ゼロ挿入シーケンスを、新しいナイキスト制限のカットオフ周波数を持つローパス フィルターに通します。このフィルターは、高周波のレプリカを減衰させ、元の周波数成分のみを保持します。
フィルタリングされた出力は、より高いサンプリングレート信号を生成し、元の信号を効果的に維持し、ダウンサンプリングプロセスを逆転させます。
フーリエ変換を使用して、-4π/9 から +4π/9 までのゼロ以外の値を示すシーケンスを考えてみましょう。
これを 2 つダウンサンプリングすると、-8π/9 から + 8π/9 までのスペクトルが得られます。
これを 8 アップサンプリングし、フーリエ変換を -π/9 から + π/9 の範囲に圧縮します。
これを 9 つダウンサンプリングし、フーリエ変換を -π から +π にスケーリングします。
アップサンプリングを 4 で組み合わせ、ダウンサンプリングを 9 で組み合わせると、エイリアシングなしで最大のダウンサンプリングが得られます。

アップサンプリング

デジタル信号処理では、信号の整合性を維持するために、信号のサンプリング 周波数を管理することが不可欠です。サンプリング周波数が低いために周波数範囲が狭くなるデシメーションされた信号は、各サンプル間にゼロを挿入することでアップサンプリングできます。このアップサンプリングプロセスでは、元のスペクトルが拡張され、新しいナイキスト周波数によって決まる間隔でスペクトルのレプリカが繰り返されます。このゼロ挿入されたシーケンスを改良するには、カットオフ周波数が新しいナイキスト限界に設定されたローパスフィルタに渡します。このフィルターは、高周波数のレプリカを減衰させ、元の周波数成分のみを保持します。

このフィルタリングプロセスの結果として、ダウンサンプリング手順を効果的に逆転させる、サンプリング周波数の高い信号が得られます。たとえば、-2π/9 から 2π/9 までのゼロ以外の値を示すフーリエ変換を持つシーケンスを考えてみましょう。このシーケンスを 4 倍にダウンサンプリングすると、スペクトルの範囲は -8π/9 から 8π/9 になります。その後、シーケンスを 2 倍にアップサンプリングすると、フーリエ変換が圧縮され、範囲は -π/9 から π/9 になります。

このアップサンプリングされたシーケンスをさらに 9 倍にダウンサンプリングすると、フーリエ変換が -2π/9 から 2π/9 に拡張されます。この 2 倍のアップサンプリングと 9 倍のダウンサンプリングの組み合わせは、9/2 倍のダウンサンプリングに相当し、エイリアシングを発生させずに最大のダウンサンプリングを実現します。

ゼロを挿入してアップサンプリングし、その後ローパスフィルタリングを行います。アップサンプリングとダウンサンプリングを正確に組み合わせるプロセスにより、信号サンプリング 周波数を効果的に管理できます。この方法により、元の信号の整合性が維持され、エイリアシングや歪みを防ぎながら、さまざまなサンプリング要件に適応できます。

このような手法は、サンプリング効率と信号忠実度のバランスが最も重要であるデジタル信号処理アプリケーションでは非常に重要です。これらのプロセスを通じてサンプリング周波数を慎重に調整することで、元の信号の本質的な特性を維持することができ、通信、オーディオ エンジニアリング、データ圧縮など、さまざまな技術分野で正確な信号処理と再構築が可能になります。

サンプリング レートが低いために周波数範囲が縮小されたデシメーション信号について考えてみましょう。
アップサンプルする各サンプル間にゼロを挿入すると、...

Education

JoVE Core

Electrical Engineering

電気工学

Unit 1

ユニット1

Sampling

サンプリング

KEY TERMS AND CONCEPTS

主要用語と概念

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...

upsampling

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 電気工学 は、電気回路の基本概念と回路解析テクニックを網羅したアニメーションビデオシリーズです。オペアンプと回路変換および 1 次・ 2 次回路の概念を網羅しています。実験を撮影した実写動画では、設計例や現代社会での応用が含まれており、複雑な概念も分かりやすく学習いただけます。

electrical engineering

HS-PS4-5

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HS-PS4-1

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HS-PS4-4

Zero-Inserted Signal Reconstruction

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