In the Discrete Fourier Transform (DFT) formula, k represents the frequency domain index. This index indicates the specific frequency component represented by a particular DFT output value.
Let's break this down further:
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DFT Overview: The DFT transforms a sequence of N complex numbers (representing a signal in the time domain) into a sequence of N complex numbers (representing the frequency domain).
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The k Index: The index 'k' ranges from 0 to N-1, where N is the length of the input sequence. Each value of 'k' corresponds to a different frequency.
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Frequency Representation: 'k' essentially specifies the harmonic frequency being represented. k = 0 corresponds to the DC component (average value), while k = 1 represents the fundamental frequency, k = 2 represents the second harmonic, and so on. Higher values of k represent higher frequencies.
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Symmetry and Interpretation: For real-valued input signals, the DFT output is symmetric around N/2. This means that the frequencies represented by k and N-k are complex conjugates of each other. Therefore, often only the first half of the DFT output (k = 0 to N/2 -1) is considered, as it contains all the unique frequency information. Sometimes, the k index is shifted to range from -N/2 to N/2-1 to more clearly represent negative and positive frequencies.
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Formula Context: The DFT formula is:
X[k] = Σ x[n] exp(-j 2π k n / N) for n = 0 to N-1
Where:
- X[k] is the k-th DFT coefficient (frequency component)
- x[n] is the n-th input sample (time-domain signal)
- N is the length of the input sequence
- j is the imaginary unit (√-1)
- k is the frequency index (0 to N-1)
- n is the time-domain index (0 to N-1)
- exp() is the exponential function
In summary, k in the DFT formula is the index that identifies a specific frequency component in the frequency domain representation of the signal. It helps to map each DFT output value to its corresponding frequency.
