When we delve into signal processing, we encounter the concept of Fourier series, which enables us to represent a signal (s(t)) through its Fourier coefficients (a_m). These coefficients are pivotal in understanding the frequency components of a signal.
Now, let's consider a scenario where we focus solely on the odd-indexed Fourier coefficients, denoted by (a_{2m+1}), to construct a new Fourier series. This selective approach results in a transformed signal, denoted as (h(t)).
Delving Deeper into the Relationship
To grasp the intricate relationship between the original signal (s(t)) and its transformed counterpart (h(t)), we look at their Fourier series representations:
The original signal is represented as: $$ s(t) = \sum_{m=-\infty}^{\infty} a_m e^{im\omega t} $$
The transformed signal, utilizing only the odd-indexed coefficients, is given by: $$ h(t) = \sum_{m=-\infty}^{\infty} a_{2m+1} e^{i(2m+1)\omega t} $$
This distinction reveals that (h(t)) is essentially crafted from the odd-indexed elements of (s(t))'s Fourier series, highlighting a specific frequency selection.
Synthesizing the Insight
By focusing exclusively on the odd-indexed Fourier coefficients to form a new series, we unveil a transformed version of the original signal. This transformation sheds light on the fundamental connection between a signal and its frequency-domain representation, specifically emphasizing the impact of selecting odd-indexed frequencies.
I welcome any further discussion, questions, or insights on this topic!