You can use wavelets in a variety of signal processing applications, such as analyzing signals at different scales, reducing noise, compressing data, and extracting features of signals. This section discusses these application areas by analyzing signals and images with the LabVIEW Wavelet Analysis Tools.

Multiscale analysis involves looking at a signal at different time and frequency scales. Wavelet transform-based multiscale analysis helps you understand both the long-term trends and the short-term variations of a signal simultaneously.

The following figure shows a multiscale analysis of a Standard & Poor's (S&P) 500 stock index during the years 1947 through 1993. The **S&P 500 Index** graph displays the monthly S&P 500 indexes. The other three graphs are the results of wavelet analysis. The **Long-Term Trend** graph is the result with a large time scale, which describes the long-term trend of the stock movement. The **Short-Term Variation** and **Medium-Term Variation** graphs describe the magnitudes of the short-term variation and medium-term variation, respectively.

Refer to the Multiscale Analysis VI in the labview\examples\Wavelet Analysis\WAApplications directory for an example of how to perform wavelet transform-based multiresolution analysis on stock indexes.

One of the most effective applications of wavelets in signal processing is denoising, or reducing noise in a signal. The wavelet transform-based method can produce much higher denoising quality than conventional methods. Furthermore, the wavelet transform-based method retains the details of a signal after denoising.

The following figure shows a signal with noise and the denoised signal using the wavelet transform-based method.

With the wavelet transform, you can reduce the noise in the signal in the **Noisy Signal** graph. The resulting signal in the **Denoised Signal** graph contains less noise and retains the details of the original signal.

Refer to the Noise Reduction VI in the labview\examples\Wavelet Analysis\WAApplications directory for an example of how to perform wavelet transform-based denoising on signals.

In many applications, storage and transmission resources limit performance. Thus, data compression has become an important topic in information theory. Usually, you can achieve compression by converting a source signal into a sparse representation, which includes a small number of nonzero values, and then encoding the sparse representation with a low bit rate. The wavelet transform, as a time-scale representation method, generates large coefficients only around discontinuities. So the wavelet transform is a useful tool to convert signals to sparse representations.

Refer to the ECG Compression VI in the labview\examples\Wavelet Analysis\WAApplications directory for an example of how to perform wavelet transform-based compression on electrocardiogram (ECG) signals.

Extracting relevant features is a key step when you analyze and interpret signals and images. Signals and images are characterized by local features, such as peaks, edges, and breakdown points. The wavelet transform-based methods are typically useful when the target features consist of rapid changes, such as the sound caused by engine knocking. Wavelet signal processing is suitable for extracting the local features of signals because wavelets are localized in both the time and frequency domains.

The following figure shows an image and the associated edge maps detected at different levels of resolutions using the wavelet transform-based method. Conventional methods process an image at a single resolution and return a binary edge map. The wavelet transform-based method processes an image at multiple levels of resolution and returns a series of grey-level edge maps at different resolutions.

A large level value corresponds to an edge map with low resolution. You can obtain the global profile of the image in a low-resolution edge map and the detailed texture of the image in a high-resolution edge map. You also can form a multiresolution edge detection method by examining the edge maps from the low resolution to the high resolution. With the multiresolution edge detection method, you can locate an object of interest in the image reliably and accurately, even under noisy conditions.

Refer to the Image Edge Detection VI in the labview\examples\Wavelet Analysis\WAApplications directory for an example of how to perform wavelet transform-based edge detection on images.