Goals For This Class

• Quickly review of the main results from last class

• Convolution and Cross-correlation

• Discrete Fourier Analysis: Important Considerations

• Some examples: How to do Fourier Analysis (IDL, MATLAB)

• Windowed Fourier Transforms and Wavelets

• Tapering

• Coherency

Time (Space) Domain Frequency Domain

From Last Class…..

Fourier Transform (Spectral Analysis)

Fourier Transform:

Inverse Fourier Transform:

The Fourier transform decomposes a function into a continuous spectrum of its frequency components (using sine and cosine functions), and the inverse transform synthesizes a function from its spectrum of frequency components

A time domain graph shows how a signal changes over time.

Frequency Vs Time (Space Domain)

A frequency domain graph shows how much of the signal lies within each given frequency band over a range of frequencies.

A frequency domain representation can also include information on the phase shift that must be applied to each sinusoid in order to be able to recombine the frequency components to recover the original time signal.

Important Properties to remember

Parseval’s Theorem

Fourier Transform conserves variance!!

Spectral Estimation

ComplexConjugate

Fourier Transform is a special case of Integral Transforms

Kernel Function

An integral transform "maps" an equation from its original "domain" to a different one.

Convolution

The convolution of two functions is defined as

Books also use…

Convolution: expresses the amount of overlap of one function as it is shifted over another function.

A convolution is a kind of very general moving average (weighted).

Convolution: Properties

Derivation:

Cross-Correlation

The cross-correlation of two functions is defined as

Relationship Between ConvolutionAnd Cross-Correlation

In General

if Spectral density

Discrete Fourier Transform

In this case we do not have a continuous function but a time series.

Time series: Sequence of data points, measured typically at successive times, separated by time intervals (often uniform).

Sampling Interval

DFT

IDFT

Discrete Fourier Transform: Properties

Fourier Transform of a real sequence of numbers results in a sequence of complex numbers of the same length.

If is real and is real

Parseval’s Theorem

Nyquist Frequency:

In order to recover all Fourier components of a periodic waveform (band-limited), it is necessary to use a sampling rate at least twice the highest waveform frequency. This implies that the Nyquist frequency is the highest frequency that can be resolved at a given sampling rate in a DFT

Sampling rate Nyquist Freq.

Similarly… Lowest Frequency?

Aliasing

Aliasing is an effect that causes different continuous signals to become indistinguishable when sampled.

Classic Example: Wagon wheels in old western movies

Good example to do in MatLab or IDL!!

Leakage

Spectral leakage appears due to the finite length of the time series (non integer number of periods, discontinuities, sampling is not and integer multiple of the period).

Allow frequency components that are not present in the original waveform to “leak” into the DFT.

How to handle this? Tapering

Matlab?

IDL?

How-To (see the code)