Frequency Domain Representations of Sampled and Wrapped Signals

Transcription

1 Frequency Domain Representations of Sampled and Wrapped Signals Peter Kabal Department of Electrical & Computer Engineering McGill University Montreal, Canada v1.5 March 2011 c 2011 Peter Kabal 2011/03/11

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3 Frequency Domain Representations of Sampled and Wrapped Signals 1 1 Introduction hese notes examine the relationships between frequency domain representations of discrete-time and wrapped signals derived from a continuous-time signal. he first part of these notes develops the relationships for periodic signals which allow for the analysis of periodic signals within the framework of the Fourier transform. he second part examines the relationships between the Fourier series, the Discrete-ime Fourier ransform (DF and the Discrete Fourier ransform (DF. hroughout this document, round brackets are used for functions of continuous variables (examples: v(t and V(ω; square brackets are used for functions of discrete variables (example: v[n]. In the first part of this document, the equations shown within boxes are results that are used in the developments leading to formulations for the Fourier transform of periodic signals. In the second part of this document, the equations shown within boxes are results which appear on the diagram relating the Fourier domain representations of sampled and wrapped signals. 2 Continuous-ime Fourier ransform he Fourier transform of a continuous-time signal is given by V(F = v(te j2πft dt. (1 his is well-defined (converges if v(t satisfies the Dirichlet conditions (absolute integrability, finite number of finite discontinuities in a finite time interval, finite number of extrema in a finite interval. he inverse transform is v(t = V(Fe j2πft df. (2 Other functions which do not satisfy the Dirichlet conditions are admissible if we allow the use of delta functions. 2.1 Dirac Delta Function he Dirac delta (impulse function can be defined in terms of its properties [1]. he sampling property of the delta function (more properly a distribution is v(0 0 A v(tδ(t dt = (3 0 0 / t A. A

4 Frequency Domain Representations of Sampled and Wrapped Signals 2 From this characterization, the delta function can be shown to be zero everywhere except at the origin, yet it has unit area, δ(t = 0 for t = 0, δ(t dt = 1. (4 he formal operations involving the delta function in an integral give us v(t t o δ(t dt = v(tδ(t+t o dt (5 and v(atδ(t dt = 1 v(tδ(t/a dt. (6 a Fourier ransforms of Delta Functions Using the sampling property of the delta function, the Fourier transform of the delta function evaluates to a constant, he inverse Fourier transform gives us δ(te j2πft dt = 1. (7 e j2πft df = δ(t. (8 his integral must be evaluated using the Cauchy principal value, i.e. as the limit /2 lim e j2πft df. (9 /2 ote that since δ(t behaves like a symmetric function, the exponent in the integral can have either sign. he inverse transform giving a delta function gives us a relation for the integral of a complex exponential. Here we restate the result using symbols which do not evoke time or frequency, e ±j2πux du = δ(x. ( Fourier Series Continuous-ime Signals A periodic function (subject to conditions of absolutely integrability over a period, a finite number of finite discontinuities in a finite interval, and a finite number of extrema in a finite interval has

5 Frequency Domain Representations of Sampled and Wrapped Signals 3 a Fourier series expansion in complex exponentials [2]. Consider a periodic function ṽ(t with period. he Fourier series expansion for ṽ(t is ṽ(t = he Fourier series coefficients are found from v m = 1 v m e j2πmt/. (11 m= /2 /2 2.3 Fourier ransform of a Continuous-ime Periodic Signal ṽ(te j2πmt/ dt. (12 We can now plunge ahead and express the Fourier transform of ṽ(t in terms of the Fourier series coefficients, V p (F = = = ṽ(te j2πft dt v m m= v m δ m= ( F m e j2πt(f m/ dt. (13 We have used Eq. (10 to evaluate the integral of the complex exponential. Summarizing, the Fourier transform of a periodic function is a sequence of delta functions in the frequency domain (at the harmonics of the periodic signal repetition rate. he areas of the delta functions are given by the Fourier series coefficients, V p (F = ( v m δ F m. (14 m= 2.4 Fourier ransform of a Periodic Impulse rain Consider the periodic impulse train (period, ṽ(t = he Fourier series coefficients for this signal are given by v m = 1 /2 /2 k= δ(t k. (15 k= ( δ t k e j2πmt/ dt. (16

6 Frequency Domain Representations of Sampled and Wrapped Signals 4 We see that the only delta function within the integration range is the one for k = 0. Using the sampling property of the delta function, the integral evaluates to unity. hen the Fourier series coefficients are constant (v m = 1/ and the Fourier transform of the impulse train is V p (F = 1 ( δ F m. (17 m= Periodic functions have delta functions in their Fourier transforms and delta functions have periodic functions in their Fourier transforms. Because of the duality between the forward and inverse Fourier transforms (if v(t V(F, then V(t v( F, this gives us the result that an impulse train (periodic and delta functions must have as its Fourier transform another impulse train (delta functions and periodic. We have an alternate formulation for the Fourier transform (or inverse Fourier transform of an impulse train. he Fourier transform for a delayed delta function δ(t k is e j2πkf. hen the Fourier transform pair is δ(t k e j2πkf. (18 k= k= An finally using the time-frequency duality of the Fourier transform, we get k= δ(t k = 1 e j2πmt/ m= k= e j2πkf = 1 ( δ F m. (19 m= 2.5 Periodic Wrapped Continuous-ime Signals Consider forming a periodic signal ṽ(t from a (non-periodic signal v(t as follows ṽ(t = v(t k= δ(t k = v(t k. (20 k= We refer to this process which forms a time-aliased periodic signal as wrapping. 1 Using the fact that a convolution in the time-domain corresponds to a product in the frequency domain, the Fourier transform of ṽ(t is k= v(t k V(F 1 ( δ F m = 1 m= ( m ( V δ F m= m. (21 1 he continuous-time signal can be considered to be wrapped onto a circle of circumference with all the superimposed intervals being added.

7 Frequency Domain Representations of Sampled and Wrapped Signals 5 his shows that the Fourier series coefficients are just V(F/ evaluated at the harmonic frequencies. Here V(F is the Fourier transform of v(t, where v(t can be longer than. We can also get the Fourier series coefficients directly from Eq. (12. We note that this equation is just a scaled version of the Fourier transform, v m = 1 V ( m, (22 where V (F is the Fourier transform of one period of ṽ(t. Given a signal v(t which is wrapped to become ṽ(t, there are two ways to get the coefficients of the frequency response of ṽ(t. he first is to take the Fourier transform of v(t (which can be longer than and then sample the frequency response at F = m/. he second is to take the Fourier transform of one period of ṽ(t and then sample the frequency response at F = m/ Poisson Sum Formula From Eq. (21, one can take the term-by-term inverse Fourier transform of the extreme righthand side expression and equate it to the lefthand side. his gives the Poisson sum formula, v(t k = 1 ( m V e k= m= j2πtm/. (23 3 Discrete-ime Fourier ransform he discrete-time Fourier transform (DF is given by V(ω = v[n]e jωn. (24 n= his sum converges if v[n] is absolutely summable. he frequency response is periodic, with period 2π. his sum can be considered to be a Fourier series expansion of the periodic signal V(ω. he inverse discrete-time Fourier transform is the computation of the Fourier series coefficients, 3.1 Fourier ransform of a Discrete-ime Periodic Signal v[n] = 1 π V(ωe jωn dω. (25 2π π Let ṽ[n] be periodic with period, ṽ[n+ ] = ṽ[n]. (26

8 Frequency Domain Representations of Sampled and Wrapped Signals 6 Let us evaluate the Fourier transform of this signal. V p ω = = = ṽ[n]e jωn n= 1 p= q=0 e p= ṽ[p + q]e jω(p+q 1 jωp q=0 ṽ[q]e jωq. (27 he second line of this equation is a result of substituting n = p + q. he third line results from exploiting the periodicity of ṽ[n]. he second factor of the result above is the DF of one period of ṽ[n]. In the last line, the first factor (sum of complex exponentials can be expressed in terms of an impulse train. he form of the sum is a little different than that encountered earlier. Appendix A recasts the earlier results in terms of radian frequency. hen from Eq. (63 with =, p= e jωp = 2π ( δ k= ω 2πk. (28 Finally we can write where V p (ω = 2π V k = 1 k= ( V k δ ω 2πk, (29 1 ṽ[n]e j2πnk/. (30 n=0 he coefficients V k are periodic with period. hey are obtained as the DF of one period of ṽ[n], evaluated at ω = 2πk/. As we will see later, these coefficients are the same as the discrete Fourier transform, except for a scale factor.

9 Frequency Domain Representations of Sampled and Wrapped Signals Fourier Series Discrete-ime Signals he Fourier series expansion for a discrete time signal can be obtained by taking the inverse transform of Eq. (29, ṽ[n] = 1 2π = = π k= π V(ωe jωn dω V k 2π ǫ ǫ 1 V k e j2πkn/. k=0 ( δ ω 2πk e jωn dω (31 In the second line, the limits of the integration have been shifted so that the delta functions for k = 0 to k = 1 fall within the limits. his is possible because the integrand is periodic with period 2π. he result is a Fourier series expansion with Fourier series coefficients V k, ṽ[n] = 3.3 Fourier ransform of a Discrete-ime Pulse rain Consider the discrete-time pulse train ṽ[n] = 1 V k e j2πkn/. (32 k=0 δ[n k]. (33 k= Here the delta function with square brackets is the unit pulse, equal to one if its argument is zero, and equal to zero otherwise. he Fourier series coefficients for this signal are constants at 1/, giving the Fourier series representation, k= δ[n k] = 1 1 e j2πnk/. (34 k=0 he DF of this pulse train can be found term-by-term for the left-hand side of the equation above, V p (ω = e jωk. (35 k=

10 Frequency Domain Representations of Sampled and Wrapped Signals 8 We can find a impulse train representation for this expression from Eq. (63 in Appendix A. his gives the following representations of a discrete-time pulse train. k= δ[n k] = 1 1 e j2πnm/ m=0 k= e jωk = 2π ( δ m= ω 2πm his expression has discrete-time pulses on the left and delta functions on the right.. ( Periodic Wrapped Discrete-ime Signals Consider forming a periodic signal ṽ[n] from v[n], ṽ[n] = v[n] k= δ[n k] = v[n k]. (37 k= Using the fact that a convolution in the time-domain corresponds to a product in the frequency domain, the DF of ṽ[n] is k= v[n k] V(ω 2π ( δ m= ω 2πm = 2π V m= ( 2πm ( δ ω 2πm. (38 Given a signal v[n] which is wrapped to become ṽ[n], there are two ways to get the coefficients of the frequency response. he first is to take the Fourier transform of v[n] and then sample the frequency response at ω = 2πm/. he second is to take the Fourier transform of one period of ṽ[n] and then sample the frequency response at ω = 2πm/. 3.5 Poisson Sum Formula For discrete-time signals we can find a result similar to the Poisson sum formula for continuoustime signals. In this case, taking the term-by-term inverse Fourier transform of the extreme righthand side of the equation above, k= v[n k] = 1 V m= ( 2πm e j2πnm/. (39

11 Frequency Domain Representations of Sampled and Wrapped Signals 9 4 Discrete Fourier ransform he discrete Fourier transform (DF for a finite length sequence x[n] is [3] V[k] = 1 v[n]e j2πnk. (40 n=0 he DF coefficients V[k] are periodic with period. he inverse discrete Fourier transform is v[n] = 1 1 V[k]e j2πnk. (41 k=0 In this equation, v[n] becomes periodic with period. With this view, we see that the DF formula Eq. (40 can be considered to operate on a finite length signal, or alternately, can be considered to operate on one period of a periodic signal. In the latter interpretation, the DF formula essentially calculates the Fourier series coefficients of the periodic signal. he DF formula differs from the Fourier series formula in Eq. (31 only in scale factor. he Fourier series coefficient V k is related to the DF coefficient as V[k] = V k. (42 5 Relationships Between the Frequency-Domain Representations he previous results put us in a good position to examine the relationships between the frequency representations of continuous-time signals, sampled signals, and periodic signals. Figure 1 shows a schematic form of the relationships. Consider the time-domain signals shown in that figure. On the left side of the diagram, the signal x[n] is formed by sampling x(t with sampling interval. he signal x[n] is formed by wrapping x[n] with period. On the right side of the diagram, the signal x(t is formed by wrapping x(t with period. Sampling x(t with period closes the loop and gives us x[n]. hus sampling then wrapping (on the left side is the same as wrapping and then sampling (on the right side with the proviso that wrapping period is times the sampling interval. 5.1 Sampling a Continuous-ime Signal: x(t x[n] We can model the sampling of a continuous-time signal as the multiplication of the continuoustime signal by a impulse train. he areas of the resulting impulses are the sample values, x s (t = x(t k= δ(t k = x(kδ(t k. (43 k=

12 Frequency Domain Representations of Sampled and Wrapped Signals m X( X c m 2 wrap xn [ ] xn ( sample Xc ( F xt ( F 1 m X m X c sample xt ( xt ( k k wrap X( DF xn [ ] xt ( FS Xm wrap xn [ ] xn [ k ] k sample 2 k Xk [ ] X xn [ ] Xk [ ] DF sample xn [ ] xn ( wrap Xk [ ] Xk m m Fig. 1 Relationships between the frequency domain representations of continuoustime signals, sampled signals, and periodic signals. F is the (continuous-time Fourier transform, DF is the discrete-time Fourier transform, FS is the Fourier series, and DF is the discrete Fourier transform.

13 Frequency Domain Representations of Sampled and Wrapped Signals 11 he Fourier transform X s (F can be computed using the relationship that a product in the time domain corresponds to a convolution in the frequency domain. x s (t = x(t δ(t k X s (F = X c (F 1 k= ( δ F m = 1 m= ( X c F m. (44 m= his frequency response is periodic with period 1/. In this form we see that sampling in the time domain corresponds to wrapping in the frequency domain. he resulting frequency response will not have aliasing (overlapping responses if the baseband signal X c (F is bandlimited to F < 1/(2. here is another expression for X s (F. his time we take the Fourier transform term by term of the second form of the time domain expression in Eq. (43, x s (t = x(kδ(t k X s (F = x(ke j2πkf. (45 k= k= In discrete-time, x[n] = x(n. (46 he DF of x[n] is X(ω which is periodic with period 2π. If we compare the definition for the DF (Eq. (24 with Eq. (45, we see that X(ω = X x (F F=ω/(2π = x[k]e jωk (47 k= he mapping between F for the continuous-time Fourier transform and ω for the discrete-time Fourier transform is ω = 2πF. With this mapping, when F increases by 1/, ω increases by 2π. as, Returning to the wrapped form of the frequency response in Eq. (44, the DF can be written X(ω = 1 m= ( ω 2πm X c. (48 2π As pointed out earlier, the x[n] X(ω relationship is that of Fourier series coefficients x[n] corresponding to a periodic signal X(ω.

14 Frequency Domain Representations of Sampled and Wrapped Signals Wrapping a Continuous-ime Signal: x(t x(t he frequency-domain consequences of wrapping a continuous-time signal have been explored in Section 2.5. hat result is reproduced here with the appropriate change of variables, x(t = x(t k 1 m ( X c( δ F m. (49 k= m= In the diagram, the frequency domain representation of the wrapped sequence is given in terms of its continuous-time Fourier series coefficients, x m = 1 m c( X. ( Wrapping a Discrete-ime Signal: x[n] x[n] he frequency-domain consequences of wrapping a discrete-time signal have been explored in Section 3.4. hat result is reproduced here with the appropriate change of variables, x[n] = k= x[n k] 2π X m= he discrete-time Fourier series coefficients for x[n] are ( 2πm ( δ ω 2πm. (51 X m = 1 ( 2πm X. (52 We can substitute for X(ω from Eq. (48 to get an expression for the Fourier series coefficients directly in terms of wrapped samples of the continuous-time Fourier transform X c (F, X k = 1 m= ( k m X c. (53 In the figure, the corresponding relationship is expressed in terms of the discrete Fourier transform coefficients (X[k] = X k, X[k] = X hese coefficients are the DF for one period of x[n]. ( 2πk. (54

15 Frequency Domain Representations of Sampled and Wrapped Signals Sampling a Continuous-ime Periodic Signal: x(t x[n] he periodic signal x(t is represented by its Fourier series coefficients x m in Eq. (50. he periodic discrete-time signal x[n] is likewise represented by its Fourier series coefficients X m in Eq. (53. he relationship between these is X k = x k m. (55 m= Finally, the DF coefficients expressed in terms of the Fourier series coefficients of x(t are given by X[k] = m= x k m. ( Frequency Domain Relationships Reversibility he diagram showing the frequency domain relationships shows that sampling in one domain corresponds to wrapping in the other domain. he diagram shows directed arrows for the sampling and wrapping operations. Under some circumstances, one can reverse the operation. For instance, sampling is reversible if a time signal is appropriately bandlimited. Similarly, wrapping a time signal is reversible if the signal is time limited to less than the wrapping period. However to reach the DF from the Fourier transform involves both sampling and wrapping. he combination is not reversible since a signal cannot be simultaneously bandlimited and time limited Periodic x(t Consider a periodic continuous-time signal x(t. Sampling this signal is well-defined. However, wrapping this signal can result in the sum becoming infinite. Since going from continuous-time to the DF input x[n] involves both sampling and wrapping, this is generally not possible for periodic x(t. Consider x(t = e j2πf ot. (57 his periodic signal a Fourier series with a single term and thus has a Fourier transform consisting of a single delta function at F = F o. Sampling x(t at k results in a DF which has a delta functions at ω = 2πF o + 2πm. here is a single delta function in any interval of length 2π. ow consider a more general periodic signal which has an unbounded number of harmonics, x(t = x m e j2πmfot. (58 m=

16 Frequency Domain Representations of Sampled and Wrapped Signals 14 If we sample this signal at k, there are two cases. If F o = M/ where M and are relatively prime, then the discrete-time signal will be periodic with period. Since the Fourier series expansion for a periodic signal with period has at most terms, the DF will contain at most delta functions in every interval of length 2π. If Fo is irrational, the DF can potentially contain an infinite number of delta functions in every interval of length 2π. he conclusion is that sampled periodic signals have a DF consisting of a finite number of delta function per 2π interval if the sampled signal is itself periodic (requires the sampling interval be synchronized with the period, or if the periodic signal has a finite number of harmonics (Fourier series expansion with a finite number of terms. 6 Summary hese notes has shown that the Fourier transform can be applied to periodic continuous-time or discrete-time signals. his allows for a unified analysis of signals containing both non-periodic and periodic components. he second part of these notes have examined the frequency domain relationships for signals derived by sampling and/or wrapping a continuous-time signal.

17 Frequency Domain Representations of Sampled and Wrapped Signals 15 Appendix A Continuous-ime Results Expressed in Radian Measure In this appendix, we restate the results derived earlier for continuous-time signals using radian frequency. Using radian frequency (Ω, the Fourier transform is he inverse transform is V(Ω = v(te jωt dt. (59 v(t = 1 V(Ωe jωt dω. (60 2π he integral representation for a delta function in Eq. (10 has a 2π factor in the exponent. Absorbing this factor into the variable u, a modified integral representation is e ±jux du = 2πδ(x. (61 Using this result, the Fourier transform of a periodic sequence expressed in terms of Ω is (c.f. Eq. (14 V p (Ω = 2π m= ( v m δ Ω 2πm, (62 where v m is given in Eq. (12. he Fourier transform of the periodic impulse train is (c.f. Eq. (19 k= δ(t k = 1 e j2πmt/ m= k= e jkω = 2π ( δ m= Ω 2πm he Poisson sum formula for the Fourier transform with radian argument is (c.f. Eq. (23 k= v(t k = 1 V m=. (63 ( 2πm e j2πtm/. (64

18 Frequency Domain Representations of Sampled and Wrapped Signals 16 References [1] A. Papoulis, he Fourier Integral and its Applications, McGraw-Hill, 1962 (ISB [2] A. V. Oppenheim and A. S. Willsky, with S. H. awab, Signals and Systems, 2nd edition, Prentice-Hall, 1996 (ISB [3] A. V. Oppenheim and R. W. Schafer, Discrete-ime Signal Processing, 3rd edition, Pearson, 2010 (ISB