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The Goertzel algorithm is a technique in digital signal processing (DSP) for efficient evaluation of the individual terms of the discrete Fourier transform (DFT). It is useful in certain practical applications, such as recognition of dual-tone multi-frequency signaling (DTMF) tones produced by the push buttons of the keypad of a traditional analog telephone.
Early uses of the term Nyquist frequency, such as those cited above, are all consistent with the definition presented in this article.Some later publications, including some respectable textbooks, call twice the signal bandwidth the Nyquist frequency; [6] [7] this is a distinctly minority usage, and the frequency at twice the signal bandwidth is otherwise commonly referred to as the Nyquist rate.
Step 2 alone creates undesirable aliasing (i.e. high-frequency signal components will copy into the lower frequency band and be mistaken for lower frequencies). Step 1, when necessary, suppresses aliasing to an acceptable level. In this application, the filter is called an anti-aliasing filter, and its design is
Fig.5: Graph of frequency aliasing, showing folding frequency and periodicity. Frequencies above f s /2 have an alias below f s /2, whose value is given by this graph. Two complex sinusoids, colored gold and cyan, that fit the same sets of real and imaginary sample points when sampled at the rate ( f s ) indicated by the grid lines.
Effects of aliasing, blurring, and sharpening may be adjusted with digital filtering implemented in software, which necessarily follows the theoretical principles. A family of sinusoids at the critical frequency, all having the same sample sequences of alternating +1 and –1.
Often, this is done by adding analog anti-aliasing filters at the input and output, thus avoiding any frequency component above the Nyquist frequency. The complexity (i.e., steepness) of such filters depends on the required signal-to-noise ratio and the ratio between the sampling rate and the highest frequency of the signal.
Instantaneous phase and frequency are important concepts in signal processing that occur in the context of the representation and analysis of time-varying functions. [1] The instantaneous phase (also known as local phase or simply phase ) of a complex-valued function s ( t ), is the real-valued function:
The Nyquist aliasing criteria is expressed graphically in the z-plane by the x-axis, where ωnT = π. The line of constant damping just described spirals in indefinitely but in sampled data systems, frequency content is aliased down to lower frequencies by integral multiples of the Nyquist frequency.