Managing signal sampling rates is essential in digital signal processing to maintain signal integrity. A decimated signal, characterized by a reduced frequency range due to its lower sampling rate, can be upsampled by inserting zeros between each sample. This upsampling process expands the original spectrum and introduces repeated spectral replicas at intervals dictated by the new Nyquist frequency. To refine this zero-inserted sequence, it is passed through a lowpass filter with a cutoff frequency set at the new Nyquist limit. This filter attenuates the higher-frequency replicas, preserving only the original frequency components.

The outcome of this filtering process is a signal with a higher sampling rate that effectively reverses the downsampling procedure. For instance, consider a sequence with a Fourier transform exhibiting non-zero values from −2π/9 to 2π/9. If this sequence is downsampled by a factor of four, its spectrum spans from −8π/9 to 8π/9. Subsequently, upsampling the sequence by a factor of two compresses the Fourier transform, now ranging from −π/9 to π/9.

Further downsampling this upsampled sequence by nine scales the Fourier transform to extend from −2π/9 to 2π/9. This combination of upsampling by two and downsampling by nine is equivalent to downsampling by a factor of 9/2, achieving the maximum downsampling without introducing aliasing.

The process of upsampling by inserting zeros and subsequent lowpass filtering, followed by precise combinations of upsampling and downsampling, allows for effective management of signal sampling rates. This method ensures that the integrity of the original signal is maintained, preventing aliasing and distortion while adapting to different sampling requirements.

Such techniques are crucial in digital signal processing applications, where the balance between sampling efficiency and signal fidelity is paramount. By carefully adjusting the sampling rates through these processes, it is possible to maintain the essential characteristics of the original signal, facilitating accurate signal processing and reconstruction in various technological domains, including communications, audio engineering, and data compression.