In generating the synchronous spectrum, it is not necessary to specify the wavelength range, why?
In the context of generating the synchronous spectrum, the wavelength range does not need to be specified because the process of generating the spectrum is based on the concept of Fourier transform, which operates on time-domain signals.
To understand this, let's first explain the concept of Fourier transform briefly. Fourier transform is a mathematical technique used to analyze signals and represent them in the frequency domain. It decomposes a time-domain signal into its constituent frequency components. In other words, it takes a signal in the time domain and converts it into a signal in the frequency domain.
Now, when we talk about the synchronous spectrum, we are usually interested in studying the frequency components of a signal at a specific point or interval within the time domain. The synchronous spectrum allows us to analyze the frequency content of a signal at a particular time or times. This is especially useful for analyzing non-stationary signals, where the frequency content of the signal may change over time.
In generating the synchronous spectrum, we apply a window function (such as a Gaussian window or a raised cosine window) to a small segment of the signal around the desired time point. This windowing operation limits the time span over which the Fourier transform is performed. By applying the Fourier transform to this windowed segment, we effectively obtain the frequency components specific to that time point or interval.
Since the synchronous spectrum is calculated for a localized section of the signal, it does not inherently require or depend on having a predefined wavelength range. The output of the synchronous spectrum analysis will typically be a representation of the frequency content in terms of amplitude or power, without explicitly specifying the corresponding wavelengths.
In summary, when generating the synchronous spectrum, we focus on a specific time point or interval within the time domain and analyze the frequency content of the signal at that point. This analysis is achieved through the application of a windowed Fourier transform, which extracts the relevant frequency components without explicitly requiring a wavelength range to be specified.