Frequency Domain Representation of Sinusoids Continuous Time Consider

Frequency Domain Representation of Sinusoids: Continuous Time Consider a sinusoid in continuous time: Frequency Domain Representation: magnitude phase radians

Example Consider a sinusoid in continuous time: Represent it graphically as: magnitude phase radians

Continuous Time and Frequency Domain In continuous time, there is a one to one correspondence between a sinusoid and its frequency domain representation: magnitude phase radians One-to-One correspondence (no ambiguity!!)

Example Let magnitude phase msec Given this sinusoid, its frequency, amplitude and phase are unique radians

Example Consider a sinusoid in discrete time: Represent it graphically as: magnitude phase radians

Frequency Domain Representation of Sinusoids: Discrete Time Same for a sinusoid in discrete time: Frequency Domain Representation: magnitude phase

Discrete Time and Frequency Domain In discrete time there is ambiguity. All these sinusoids have the samples: with k integer

Example All these sinusoids have the samples: … and many more!!!

Ambiguity in the Digital Frequency The given sinusoid can come from any of these frequencies, and many more!

In Summary A sinusoid with frequency is indistinguishable from sinusoids with frequencies These frequencies are called aliases.

Where are the Aliases? Notice that, if the digital frequency is in the interval all its aliases are outside this interval … … …all aliases here…

Discrete Time and Frequency Domains If we restrict the digital frequencies within the interval there is a one to one correspondence between sampled sinusoids and frequency domain representation (no aliases) magnitude phase

Continuous Time to Discrete Time Now see what happens when you sample a sinusoid: how do we relate analog and digital frequencies?

Which Frequencies give Aliasing? … … Aliases: … k integer

Example Given: a sinusoid with frequency sampling frequency the aliases (ie sinusoids with the samples as the one given) have frequencies

Example

Aliased Frequencies aliases

Sampling Theorem for Sinusoids If you sample a sinusoid with frequency such that , there is no loss of information (ie you reconstruct the same sinusoid) magnitude DAC Digital to Analog Converter

Extension to General Signals: the Fourier Series Any periodic signals with period can be expanded in a sum of complex exponentials (the Fourier Series) of the form with the fundamental frequency The Fourier Coefficients

Example A sinusoid with period We saw that we can write it in terms of complex exponentials as Which is a Fourier Series with

Computation of Fourier Coefficients For general signals we need a way of determining an expression for the Fourier Coefficients. From the Fourier Series multiply both sides by a complex exponential and integrate

Fourier Series and Fourier Coefficients Fourier Series: Fourier Coefficients:

Example of Fourier Series… Period Fourier Coefficients: Fundamental Frequency:

… Plot the Coefficients Fourier Coefficients:

Parseval’s theorem The Fourier Series coefficients are related to the average power as

Sampling Theorem If a signal is a sum of sinusoids and B is the maximum frequency (the Bandwidth) you can sample it at a sampling frequency without loss of information (ie you get the same signal back) magnitude DAC Digital to Analog Converter

Example it has two frequencies The bandwidth is The sampling frequency has to be so that we can sample it without loss of information

Example The bandwidth of a Hi Fidelity audio signal is approximately since we cannot hear above this frequency. The music on the Compact Disk is sampled at i. e. 44, 100 samples for every second of music

Example For an audio signal of telephone quality we need only the frequencies up to 4 k. Hz. The sampling frequency on digital phones is