The Fourier Series for Discrete-Time Signals

1. The Fourier Series for Discrete-Time Signals Suppose that we are given a periodic sequence with period N. The Fourier series representation for x[n] consists of N harmonically related exponential functions ej2kn/N, k = 0, 1,2,…….,N-1 and is expressed as where the coefficients ck can be computed as:

2. Example 2:Determine the spectra of the following signals:(a) x[n] = [1, 1, 0, 0], x[n] is periodic with period 4 (b) x[n] = cosn/3(c) x[n] = cos(2)n Solution: (a) x[n] = [1, 1, 0, 0] Now

3. The magnitude spectra are: and the phase spectra are:

4. (b) x[n] = cosn/3 Solution: In this case, f0 = 1/6 and hence x[n] is periodic with fundamental period N = 6. Now Similarly, c2 = c3 = c4 = 0, c1 = c5 = ½.

5. (c) Cos(2)n Solution: The frequency f0 of the signal is 1/2 Hz. Since f0 is not a rational number, the signal is not periodic. Cosequently, this signal cannot be expanded in a Fourier series.

6. Power density Spectrum of Periodic Signals The average power of a discrete time periodic signal with period N is The above relation may also be written as or This is Parseval’s Theorem for Discrete-Time Power Signals.

7. Similarly, for discrete time energy signals, the Parseval’s Theorem may be stated as follows: If the signal x[n] is real, [i.e. x*[n] = x[n]], then we can easily show that |c-k| = |ck| (even symmetry) -c-k = ck (odd symmetry) |ck| = |cN-k| (Periodicity) ck = cN-k (periodicity)

8. More specifically, we have |c0| = |cN| c0 = - cN |c1| = |cN-1| c1 = - cN-1 |cN/2| = |cN/2| cN/2 = 0 if N is even |c(N-1)/2| = |c(N+1)/2| c(N-1)/2 = (N+1)/2 if N is odd

9. Example: Determine the Fourier Series Coefficients and the Power Density Spectrum of the following periodic signal. X[n] A Solution: n -N N L k = 0, 1, 2, …., N-1

10. But Therefore,

11. The Fourier Transform of Discrete-Time Aperiodic Signals The Fourier Transform of a finite energy discrete time signal x[n] is defined as X(w) may be regarded as a decomposition of x[n] into its Frequency components. It is not difficult to Verify that X(w) is periodic with frequency 2. The Inverse Fourier Transform of X(w) may be defined as

12. Energy Density Spectrum of Aperiodic Signals Energy of a discrete time signal x[n] is defined as Let us now express the energy Ex in terms of the spectral characteristic X(w). First we have If we interchange the order of integration and summation in the above equation, we obtain

13. Therefore, the energy relation between x[n] and X(w) is This is Parseval’s relation for discrete-time aperiodic signals.

14. Example: Determine and sketch the energy density spectrum of the signal x[n] = anu[n], -1<a<1 Solution: The energy density spectrum (ESD) is given by X(w) a = 0.5 a= -0.5 w   0

15. Example: Determine the Fourier Transform and the energy density spectrum of the sequence Solution: The magnitude of x[n] is and the phase spectrum is The signal x[n] and its magnitude is plotted on the next slide. The Phase spectrum is left as an exercise.

16. x[n] |X(w)|

17. Properties of Discrete Time Fourier Transform (DTFT) Symmetry Properties: Suppose that both the signal x[n] and its transform X(w) are complex valued. Then x[n] = xR[n] + jxI[n] (1) X(w) = XR(w) + jXI[w] (2) Substitution of (1) and (2) gives Separating the real and imaginary parts, we have