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Discrete Time Periodic Signals A discrete time signal x[n] is periodic with period N if and only if for all n. Definition: Meaning: a periodic signal keeps repeating itself forever!

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Example: a Sinusoid Consider the Sinusoid: It is periodic with period since for all n.

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General Periodic Sinusoid Consider a Sinusoid of the form: It is periodic with period N since for all n. with k, N integers.

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Consider the sinusoid: It is periodic with period since for all n. We can write it as: Example of Periodic Sinusoid

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Consider a Complex Exponential of the form: for all n. It is periodic with period N since Periodic Complex Exponentials

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Consider the Complex Exponential: We can write it as Example of a Periodic Complex Exponential and it is periodic with period N = 20.

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Goal: We want to write all discrete time periodic signals in terms of a common set of “reference signals”. Reference Frames It is like the problem of representing a vector in a reference frame defined by an origin “0” reference vectors Reference Frame

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Reference Frames in the Plane and in Space For example in the plane we need two reference vectors Reference Frame … while in space we need three reference vectors Reference Frame

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A Reference Frame in the Plane If the reference vectors have unit length and they are perpendicular (orthogonal) to each other, then it is very simple: Where projection of along projection of along The plane is a 2 dimensional space.

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A Reference Frame in the Space If the reference vectors have unit length and they are perpendicular (orthogonal) to each other, then it is very simple: Where projection of along projection of along The “space” is a 3 dimensional space.

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Example: where am I ? N E Point “x” is 300m East and 200m North of point “0”.

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Reference Frames for Signals We want to expand a generic signal into the sum of reference signals. The reference signals can be, for example, sinusoids or complex exponentials reference signals

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Back to Periodic Signals A periodic signal x[n] with period N can be expanded in terms of N complex exponentials as

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A Simple Example Take the periodic signal x[n] shown below: Notice that it is periodic with period N=2. Then the reference signals are We can easily verify that (try to believe!): for all n.

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Another Simple Example Take another periodic signal x[n] with the same period (N=2): Then the reference signals are the same We can easily verify that (again try to believe!): for all n. Same reference signals, just different coefficients

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Orthogonal Reference Signals Notice that, given any N, the reference signals are all orthogonal to each other, in the sense Since by the geometric sum

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… apply it to the signal representation … and we can compute the coefficients. Call then

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Discrete Fourier Series Given a periodic signal x[n] with period N we define the Discrete Fourier Series (DFS) as Since x[n] is periodic, we can sum over any period. The general definition of Discrete Fourier Series (DFS) is for any

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Inverse Discrete Fourier Series The inverse operation is called Inverse Discrete Fourier Series (IDFS), defined as

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Revisit the Simple Example Recall the periodic signal x[n] shown below, with period N=2: Then Therefore we can write the sequence as

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Example of Discrete Fourier Series Consider this periodic signal The period is N=10. We compute the Discrete Fourier Series

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… now plot the values …

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Example of DFS Compute the DFS of the periodic signal Compute a few values of the sequence and we see the period is N=2. Then which yields

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Signals of Finite Length All signals we collect in experiments have finite length Example: we have 30ms of data sampled at 20kHz (ie 20,000 samples/sec). Then we have

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Series Expansion of Finite Data We want to determine a series expansion of a data set of length N. Very easy: just look at the data as one period of a periodic sequence with period N and use the DFS:

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Discrete Fourier Transform (DFT) Given a finite interval of a data set of length N, we define the Discrete Fourier Transform (DFT) with the same expression as the Discrete Fourier Series (DFS): And its inverse

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Signals of Finite Length All signals we collect in experiments have finite length in time Example: we have 30ms of data sampled at 20kHz (ie 20,000 samples/sec). Then we have

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Series Expansion of Finite Data We want to determine a series expansion of a data set of length N. Very easy: just look at the data as one period of a periodic sequence with period N and use the DFS:

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Discrete Fourier Transform (DFT) Given a finite of a data set of length N we define the Discrete Fourier Transform (DFT) with the same expression as the Discrete Fourier Series (DFS): and its inverse

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Example of Discrete Fourier Transform Consider this signal The length is N=10. We compute the Discrete Fourier Transform

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… now plot the values …

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DFT of a Complex Exponential Consider a complex exponential of frequency rad. We take a finite data length … and its DFT How does it look like?

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Recall Magnitude, Frequency and Phase 2. We represented it in terms of magnitude and phase: magnitude phase Recall the following: 1. We assume the frequency to be in the interval

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Compute the DFT… Notice that it has a general form: where (use the geometric series)

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See its general form:

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… since:

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… and plot the magnitude

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Example Consider the sequence In this case Then its DFT becomes Let’s plot its magnitude:

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... first plot this …

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… and then see the plot of its DFT The max corresponds to frequency

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Same Example in Matlab Generate the data: >> n=0:31; >>x=exp(j*0.3*pi*n); Compute the DFT (use the “Fast” Fourier Transform, FFT): >> X=fft(x); Plot its magnitude: >> plot(abs(X)) … and obtain the plot we saw in the previous slide.

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Same Example in Matlab Generate the data: >> n=0:31; >>x=exp(j*0.3*pi*n); Compute the DFT (use the “Fast” Fourier Transform, FFT): >> X=fft(x); Plot its magnitude: >> plot(abs(X)) … and obtain the plot we saw in the previous slide.

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Same Example (more data points) Consider the sequence In this case >> n=0:255; >>x=exp(j*0.3*pi*n); >> X=fft(x); >> plot(abs(X)) See the plot …

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… and its magnitude plot

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What does it mean? The max corresponds to frequency A peak at index means that you have a frequency

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Example You take the FFT of a signal and you get this magnitude: There are two peaks corresponding to two frequencies:

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DFT of a Sinusoid Consider a sinusoid with frequency rad. We take a finite data length … and its DFT How does it look like?

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Sinusoid = sum of two exponentials Recall that a sinusoid is the sum of two complex exponentials magnitude phase

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Use of positive frequencies Then the DFT of a sinusoid has two components … but we have seen that the frequencies we compute are positive. Therefore we replace the last exponential as follows:

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Represent a sinusoid with positive freq. Then the DFT of a sinusoid has two components magnitude phase

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Example Consider the sequence In this case Then its DFT becomes Let’s plot its magnitude:

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... first plot this …

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… and then see the plot of its DFT The first max corresponds to frequency This is NOT a frequency

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Symmetry If the signal is real, then its DFT has a symmetry: In other words: Then the second half of the spectrum is redundant (it does not contain new information)

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Back to the Example: If the signal is real we just need the first half of the spectrum, since the second half is redundant.

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Plot half the spectrum If the signal is real we just need the first half of the spectrum, since the second half is redundant.

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Same Example in Matlab Generate the data: >> n=0:31; >>x=cos(0.3*pi*n); Compute the DFT (use the “Fast” Fourier Transform, FFT): >> X=fft(x); Plot its magnitude: >> plot(abs(X)) … and obtain the plot we saw in the previous slide.

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Same Example (more data points) Consider the sequence In this case >> n=0:255; >>x=cos(0.3*pi*n); >> X=fft(x); >> plot(abs(X)) See the plot …

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… and its magnitude plot The first max corresponds to frequency

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Example You take the FFT of a signal and you get this magnitude: There are two peaks corresponding to two frequencies:

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