Wavelet Transform Example: Analyzing f(t) = sin(2π×500t) + sin(2π×1000t) + 1.5δ(t-165) + 1.5δ(t-207)

Using the example discussed in the text, we examine the wavelet transform application for the function f(t) = sin(2π×500t) + sin(2π×1000t) + 1.5δ(t-165) + 1.5δ(t-207). Through wavelet transformation, we can convert this function's time-domain representation into a frequency-domain representation. This approach enables comprehensive analysis of signal frequency components and temporal localization properties. The implementation typically involves selecting an appropriate wavelet family (such as Morlet or Daubechies wavelets) and applying discrete wavelet transform (DWT) or continuous wavelet transform (CWT) algorithms. Key steps include signal preprocessing, wavelet coefficient computation using convolution operations, and time-frequency visualization through scalogram plots. The delta functions at t=165 and t=207 will appear as localized energy concentrations in the time-frequency plane, while the sinusoidal components will show distinct frequency bands at 500Hz and 1000Hz throughout the signal duration. This analysis reveals both stationary frequency content and transient events within the composite signal.