Normalized frequency (signal processing)
Normalized frequency, in the context of signal processing, is a way to express the frequency of a signal relative to a reference frequency, often the sampling rate of a discrete-time system. Instead of expressing frequency in Hertz (Hz) or radians per second, it is represented as a fraction or multiple of the sampling rate. This provides a scale-invariant representation, useful when comparing or analyzing signals sampled at different rates.
There are two common forms of normalized frequency:
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Frequency normalized by the Nyquist rate: In this representation, the frequency is normalized to the Nyquist rate, which is half the sampling rate. This results in a normalized frequency range of -1 to 1, or 0 to 1. A normalized frequency of 1 (or -1) corresponds to the Nyquist rate, meaning the signal oscillates at the highest frequency that can be accurately represented given the sampling rate. This form is often expressed as f/fN, where f is the frequency in Hertz and fN is the Nyquist frequency. This representation is particularly useful for visualizing spectra or analyzing digital filters.
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Frequency normalized by the sampling rate: In this representation, the frequency is normalized by the sampling rate itself. This results in a normalized frequency range of -0.5 to 0.5, or 0 to 1. A normalized frequency of 1 corresponds to the sampling rate. This form is often expressed as f/fs, where f is the frequency in Hertz and fs is the sampling frequency. Sometimes it is represented in radians as ω/ωs where ω is the angular frequency in radians per second, and ωs is the angular sampling frequency in radians per second.
The primary advantage of using normalized frequency is its independence from specific sampling rates. This allows for easier comparisons and implementations of algorithms across different systems. For example, the coefficients of a digital filter designed with normalized frequency parameters will work correctly regardless of the actual sampling rate used, as long as the input signal's frequency content is properly scaled relative to the sampling rate. Furthermore, it simplifies theoretical analysis and simulations, as the frequency range is inherently bounded between manageable values.