Frequency Domain Representation of Sinusoids: Continuous Time

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

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

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

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

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

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

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

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

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

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

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

12. magnitude phase 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)

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

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

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

16. Example

17. Aliased Frequencies aliases

18. 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

19. 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

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

21. 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

22. Fourier Series and Fourier Coefficients Fourier Series: Fourier Coefficients:

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

24. … Plot the Coefficients Fourier Coefficients:

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

26. magnitude DAC Digital to Analog Converter 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)

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

28. 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

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