16.4:
Properties of Fourier series II
When a signal undergoes time scaling, the Fourier series coefficients remain the same, but the representation of the Fourier series changes due to an alteration in the fundamental frequency.
Function symmetries, even, odd, or half-wave odd, are also essential for understanding the Fourier series.
A function is considered even if it remains the same when its input is negated. If a function is even, the Fourier series is simplified as specific components vanish.
A function is deemed odd if it changes sign when its input is negated. If a function is odd, again, the Fourier series simplifies as other components disappear.
A function exhibits half-wave symmetry if it looks identical to its original form when shifted by half a period and inverted. Interestingly, the Fourier series of a half-wave symmetric function contains only odd harmonics.
In music production and multimedia, time scaling adjusts the speed of audio signals, allowing for pitch correction and playback control.
In image processing, even-odd symmetry aids in efficient image reconstruction and compression, leading to optimal storage and improved visualization.
16.4:
Properties of Fourier series II
Time scaling of signals is a crucial concept in signal processing that affects the Fourier series representation without altering its coefficients. The process modifies the fundamental frequency, thereby changing how the series represents the signal over time. This principle is essential in various applications, including audio and image processing, where signal manipulation is frequent. Understanding function symmetries is fundamental to simplifying the Fourier series.
A function f(t) is considered even if f(t) = f(−t). For even functions, the Fourier series simplifies because all sine terms, which are odd functions, vanish. This reduction occurs because the integral of an odd function over a symmetric interval around zero is zero.
A function f(t) is deemed odd if f(t) = −f(−t). For odd functions, the Fourier series simplifies differently; all cosine terms, which are even functions, disappear. This is due to the same principle that the integral of an odd function over a symmetric interval is zero.
A function exhibits half-wave symmetry if f(t+T/2) = −f(t), where T is the period of the function. For functions with half-wave symmetry, the Fourier series contains only odd harmonics. This means that the series is composed solely of terms with frequencies that are odd multiples of the fundamental frequency, further simplifying the series representation.
The implications of time scaling and function symmetries are profound in practical applications. In music production, time scaling is used to adjust the playback speed of audio signals. This technique is essential for pitch correction, allowing audio engineers to modify the speed without altering the pitch or vice versa. It enables precise control over audio playback, ensuring high-quality sound reproduction.
Even-odd symmetry properties are leveraged for efficient image reconstruction and compression. By recognizing and utilizing these symmetries, algorithms can reduce the amount of data needed to represent an image, leading to optimal storage solutions and improved visualization. Symmetrical properties help in achieving higher compression ratios without compromising image quality.