ECE-700 Review. Phil Schniter. January 5, x c (t)e jωt dt, x[n]z n, Denoting a transform pair by x[n] X(z), some useful properties are

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1 ECE-7 Review Phil Schniter January 5, 7 ransforms Using x c (t) to denote a continuous-time signal at time t R, Laplace ransform: X c (s) x c (t)e st dt, s C Continuous-ime Fourier ransform (CF): ote that: X c (j) x c (t) x c (t)e jt dt, X c (j)e jt d, R t R X c (j) is the Laplace transform evaluated at s j. x c (t) R implies X c (j) Xc ( j), i.e., conjugate symmetry. x c (t) is bandlimited to if X c (j) for all >. Using x[n] to denote a discrete-time signal at index n Z, z-transform: X(z) x[n]z n, z C Denoting a transform pair by x[n] X(z), some useful properties are x[ n] X(z ) ( ) n x[n] X( z) c P. Schniter,

2 Discrete-ime Fourier ransform (DF): ote that: X(e jω ) x[n] x[n]e jωn, ω R X(e jω )e jωn dω, n Z X(e jω ) is the z-transform evaluated on the unit circle in C-plane: z e jω. X(e jω ) is -periodic in ω. x[n] R implies X(e jω ) X (e jω ), i.e., conjugate symmetry. Other DF properties are: x[ n] X(e jω ) x [n] X (e jω ) x[n l] X(e jω )e jωl x[n]e jω n X(e j(ω ω) ) x[n]y[n] X(ejθ )Y (e j(ω θ) )dθ x[n] y[n] X(e jω )Y (e jω ) n x[n] X(ejω ) dω where denotes linear convolution: x[n] y[n] m x[m] y[n m]. Uniform Sampling Say that x[n] is sampled from x c (t) with uniform sampling interval : x[n] x c (n), n Z Let us define the continuous-time impulse train p(t) δ(t n) where δ(t) denotes the Dirac delta function, defined by the properties δ(t)dt unit area f(t)δ(t τ)dt f(τ) sifting property Using Fourier series, it can be shown that p(t) k e j kt c P. Schniter,

3 x c (t) x s (t) t x[n] t Figure : Signals used in sampling theorem. n Multiplying x c (t) by the impulse train yields the continuous-time sampled signal x s (t) which will help us to derive the sampling theorem. (See Fig..) aking the CF of x s (t), X s (j) x s (t) x c (t) x c (t) ( k k x c (t) k δ(t n) k e j kt e j kt )e jt dt x c (t)e j( k)t dt X c ( j( k)) () otice that X s (j) is a summation of scaled and shifted copies of X c (j). When X c (j) is bandlimited to rad/s the spectral copies do not overlap, and we say there is no c P. Schniter, 3

4 aliasing. Aliasing may result when X c (j) is not bandlimited to. (See Fig..) he frequency rad/s, i.e., Hz, is often called the yquist frequency. X c (j) X c (j) X s (j) / images X s (j) / images Figure : Example of no aliasing (left) and aliasing (right) in X s (j). We now investigate the relationship between the CF and the DF: X(e jω ) x[n]e jωn x c (n)e jωn ( x c (t) x c (t)δ(t n)dt ) e jωn x c (t)δ(t n)e jωn dt }{{} nonzero iff n t/. x c (t)δ(t n)e jω t dt δ(t n) e j ω t dt }{{} x s (t) ( X s j ω ) () c P. Schniter, 4

5 Plugging () into () yields X(e jω ) k ( ω k ) X c (j ) otice that the DF of the sampled signal x[n] is a summation of scaled, stretched, and shifted copies of the CF of the continuous signal x c (t). As implied by (), the DF may show evidence of aliasing when x c (t) is not bandlimited to the yquist frequency. (See Fig. 3.) X c (j ω ) X c (j ω ) ω ω X(e jω ) / images X(e jω ) / images ω ω Figure 3: Example of no aliasing (left) and aliasing (right) in X(e jω ). 3 Reconstruction Fig. 4 illustrates the theoretical procedure by which a yquist-bandlimited x c (t) can be perfectly reconstructed from its sampled representation x[n]. Essentially, it is a reversal of the sampling process. First, the discrete-time sequence x[n] is convolved with a continuous-time Dirac delta to obtain x s (t). x s (t) x[n]δ(t n) As we know, X s (j) contains unwanted spectral copies, or images, and is scaled by a factor of / relative to X c (j). hus, the second step in the reconstruction procedure is to remove these images and scale by. his can be accomplished by a brick-wall analog lowpass filter with cutoff at and DC gain of. he time-domain impulse response of this filter is given by h b (t) H b (j)e jt d / e jt d sin(t/) / t/ c P. Schniter, 5

6 See Fig. 5 for an illustration. hus, the reconstructed signal can be written x c (t) x s (t) h b (t) x[n] sin( (t n)) (t n) (3) x[n] δ(t) x s (t) h b (t) sin(t/) t/ x c (t) x[n] x s (t) x c (t) n t t X(e jω ) X s (j) X c (j) / / ω Figure 4: Ideal reconstruction of yquist-bandlimited signal. H b (j) h b (t) sin(t/) t/ t Figure 5: Brick-wall analog reconstruction lowpass filter. c P. Schniter, 6

7 In practice, it is not possible to generate Dirac deltas for the creation of x s (t). So, instead of convolving with δ(t), we might convolve with a rectangular pulse of width, known as a zero-order hold (ZOH) function and denoted by h z (t). his yields x z (t): x z (t) x[n]h z (t n) Unwanted spectral copies in X z (j) can be removed by a final stage of lowpass filtering. he two-step procedure is illustrated in Fig. 6. o be consistent with the Dirac delta, we assume a ZOH function with unit area, requiring that the analog reconstruction lowpass filter has DC gain. x[n] ZOH DAC h z(t) x z (t) reconstruction filter h r (t) x c (t) x[n] x z (t) x c (t) n t t X(e jω ) X droop z (j) X c (j) / / Figure 6: ZOH reconstruction of yquist-bandlimited signal. Unlike convolution with δ(t), convolution with the ZOH function h z (t) introduces passband droop and out-of-band attenuation. his results from the fact that the frequency response magnitude of the ZOH function is not constant in : H z (j) h z (t)e jt dt e jt dt sin(/) / e j/ (See Fig. 7 for an illustration.) hus, for perfect reconstruction, the analog reconstruction filter H r (j) must invert the ZOH response H z (j) over the passband [, ). he left side of Fig. 8 shows the ideal H r (j). c P. Schniter, 7

8 / h z (t) H z (j) t 4 4 Figure 7: ZOH function. It is difficult to build analog reconstruction filters with sharp cutoff. Instead, one hopes that the desired signal is bandlimited to less than the yquist frequency (as in Fig. 6), so that there is an absence of unwanted spectral energy in a region around ±/. In this case, the analog reconstruction filter H r (j) may be designed with a wider transition band, such as on the right side of Fig. 8. H r (j) H r (j) Figure 8: Ideal (left) and practical (right) reconstruction filters for ZOH. c P. Schniter, 8

9 4 Discrete Fourier ransform -point Discrete-Fourier ransform (DF): X[k] x[n] n x[n]e j kn, k... X[k]e j kn, n... k ote that: X[k] is the DF evaluated at ω k for k.... Zero-padding x[n] to M samples (where M > ) prior to an M-pt DF yields an M-point uniformly sampled version of the DF: X(e j M k ) n x[n]e j M kn M n x zp [n]e j M kn X zp [k], k...m his can be used to compute a densely sampled DF of any -pt sequence. An -pt DF can be interpolated to reconstruct the DF of an -pt sequence. X(e jω ) n n x[n]e jωn X[k]e j kn e jωn k X[k] k X[k] k n e j(ω k)n sin ( ω k sin ( ω k ) ) e j(ω k) sin ( ( sin ω ω ) ) 4 Figure 9: Dirichlet sinc c P. Schniter, 9

10 he DF has a convenient matrix representation. X[] e j e j e j e j... x[] X[] e j e j e j e j 3... x[]. e j e j e j 4 e j X[ ] x[ ] }{{}}{{}}{{} X W x where W has the following properties: [W] k,n e j kn W is symmetric, i.e., W W t. W is unitary, i.e., ( W)( W) H ( W) H ( W) I. W W, where W is called the IDF matrix. W is a Vandermonde matrix, i.e., the n th column of W is formed by raising e j n to the powers k,,...,. he Vandermonde property implies that W is full rank. For m with m, the FF algorithm can be used to compute the DF using approximately log, rather than, operations. 5 Miscellaneous savings factor Linear Phase: A linear-phase filter H(z) has a phase response that is linear in frequency. Specifically, the DF of a linear phase filter can be written as { H(e jω ) e jωd H(e jω d R ), H(e jω ) R When d, we refer to the filter as zero-phase. It can be shown that a filter is zero-phase iff its impulse response is conjugate symmetric around the origin, i.e., h[m] h [ m] m. (See the proof below.) Since symmetric length- causal filters exhibit coefficient symmetry around the index, the same arguments can be used to show that such filters are linear phase with d. c P. Schniter,

11 Proof. If H(e jω ) is real-valued, then following must hold ω: { } Im h[n]e jnω Im { h i [] + If f(ω) ω, then ( hr [n] + jh i [n] )( cos(nω) j sin(nω) )} h i [n] cos(nω) h r [n] sin(nω) ( hi [n] + h i [ n] ) cos(nω) ( h r [n] h r [ n] ) sin(nω) n lim ω ω Using f(ω) Im{H(e jω )}, we find that Similarly, if f(ω) ω, then lim ω ω ω f(ω)sin(ωm)dω m h r [m] h r [ m] m. ω Using f(ω) Im{H(e jω )}, we find that ω ω f(ω)cos(ωm)dω m h i [m] + h i [ m] m Z +. Putting the real/imaginary coefficient requirements together, we have h[m] h [ m] m Z +. hus we have shown that a real-valued DF implies conjugate symmetric coefficients (with symmetry around the origin). o prove the other direction, assume h[m] h [ m] m Z +. hen H(e jω ) h[] + h[] + h[n]e jnω h[n]e jnω + h[ n]e jnω n (h[n]e jnω ) + (h[n]e jnω ) n { } h[] + Re h[n]e jnω R ω since the conjugate symmetry implies that h[] R. hus we have shown that conjugate symmetric coefficients (with symmetry around the origin) imply a real-valued DF. n c P. Schniter,

12 Group-Delay Response: he group-delay response of a discrete-time linear system H(z) is defined as the negative derivative of the phase response of the DF H(e jw ). In other words, if we write H(e jω ) H(e jω ) e jφ(ω) where φ(ω) denotes the phase response, then the group-delay response g(ω) is g(ω) φ(ω) ω Functionally, g(ω) describes the delay (in samples) imposed by H(z) on input signal components with frequency ω. Recall that the DF of a symmetric causal length- filter can be written in terms of real-valued H(e jω ) as H(e jω ) H(e jω j )e ω from which it is evident that H(e jω ) H(e jω ) sgn{ H(e jω )} H(e jω ). hus H(e jω ) H(e jω ) sgn{ H(e jω j )}e ω which implies that φ(ω) g(ω) { ω ω s.t. sgn{ H(e jω )} ω + ω s.t. sgn{ H(e jω )} { ω s.t. H(e jω ) undefined ω s.t. H(e jω ) ote that this group-delay response is constant with respect to frequency, implying that all input components not completely attenuated by the system will be delayed by the same amount. c P. Schniter,