X. Chen More on Sampling

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1 X. Chen More on Sampling 9 More on Sampling 9.1 Notations denotes the sampling time in second. Ω s = 2π/ and Ω s /2 are, respectively, the sampling frequency and Nyquist frequency in rad/sec. Ω and ω denote, respectively, frequency in rad/s (used in continuoustime signal analysis), and frequency in rad (for discretetime signal processing). We have ω = Ω (12) We use square bracket, e.g. x [n], to indicate a discrete sequence; and regular parenthesis, e.g. x c (t), to indicate a continuoustime signal. X c (jω) = F{x c (t)} ˆ x c (t)e jωt dt denotes the Fourier transform of a continuoustime signal x c (t); X(e jω ) = F d {x[n]} x [n] e jωn denotes the discretetime Fourier transform of a discrete sequence x [n] = x c (n ). S denotes the sampler that samples a continuoustime signal to discretetime sequences. H denotes a holder performing, e.g. zero order hold, to transform a discretetime sequence to a continuoustime signal. 42 version September 23, 2015

2 X. Chen Impulse and impulse trains 9.2 Impulse and impulse trains (Spectrum of an impulse) The Fourier transform of a shifted delta function is F (δ (t a)) = e jωa. In other words, 5 δ (t a) = F ( 1 e jωa) = 1 ˆ e jωa e jωt dω (13) 2π Understanding the result: Recall that in Laplace transform, shifted impulse in time corresponds to delay in frequency domain. (Spectrum of sinusoidals) The Fourier transform of a singlefrequency complex sinusoidal signal x c (t) = e jω 0t is F ( e jω 0t ) = 2πδ (Ω Ω 0 ) [by using (13)]. Real signals have conjugate symmetric spectra. In other words, if F (jω) = ˆ f (t) e jωt dt then F (jω) = ˆ f (t) e jωt dt = F ( jω) In particular, the Fourier transform of a real sinusoidal signal x c (t) = cos (Ω 0 t) = ( e jω 0t + e jω 0t ) /2 is F (cos (Ω 0 t)) = πδ (Ω Ω 0 ) + πδ (Ω + Ω 0 ) (14) 5 The equality also provides an alternative definition of the delta function: δ (t) = 1 ˆ e jωt dω 2π 43 version September 23, 2015

3 X. Chen C/D process 9.3 C/D process The C/D process from a continuous signal x c (t) to the discrete sampled sequence x [n] = x c (n ), namely x c (t) S x [n] can be mathematically represented as Fig. 6, where s (t) = δ (t n) x c (t) Multiplication x s(t) Impulse train to discretetime sequence x [n] Figure 6 Mathematical representation of the C/D process x s (t) = x c (t) s (t) = x c (t) δ (t n ) = x c (n ) δ (t n ) (15) 44 version September 23, 2015

4 X. Chen C/D process Continuoustime signal to continuoustime impulse train By Fourier series expansion, 6 the continuoustime impulse train contains infinite amount of frequency components: δ (t k ) = 1 e jωskt The Fourier transformation of the impulse train in Fig. 6 is thus S (jω) = F (s (t)) = 2π δ (Ω kω s ) = 2π where Ω s = 2π/ is the sampling frequency in rad/sec. ( δ Ω k 2π ) By convolution property, the Fourier transform of x s (t) in (15) is X s (jω) = F (x c (t) s (t)) = 1 2π X c (jω) S (jω) = 1 X c (j (Ω kω s )) (16) Hence if the spectrum of x c (t) contains components beyond the Nyquist frequency Ω s /2, aliasing will occur in obtaining the spectrum of x s (t). example: consider sampling x c (t) = cos (Ω 0 t) and x c (t) = cos ((Ω 0 + Ω s ) t) at sampling frequency Ω s, with Ω 0 < Ω s /2. 6 If f(t) is periodic with period, then f(t) = 1 < f(x), e jωskt > e jωskt where ω s = 2π/ and < f(x), e jωskt > is the inner product defined by < f(x), e jωskt >= ˆ Ts/2 /2 f(x)e jωskt dt 45 version September 23, 2015

5 X. Chen C/D process x [n] will have a spectrum of periodic spikes with base pattern at Ω 0 and Ω 0 [recall (14)] x [n] will have a spectrum of periodic spikes with base pattern at Ω 0 and Ω 0 as well [recall (16)]! Example: if x c (t) = e jωot, then x s (t) = δ (t n ) e jωot (17) and X s (jω) = 1 Hence by inverse Fourier transform x s (t) = 1 which is an equivalent form of (17). δ (j (Ω kω s )) e j(ωo+ωsn)t Continuoustime impulse train to discretetime sequence The discretetime Fourier transform of x [n] = x c (nt ) is X ( e jω) = x [n] e jωn Directly taking the Fourier transfrm of (15) yields ( ) X s (jω) = F x c (n ) δ (t n ) = x (nt ) e jωtsn 46 version September 23, 2015

6 X. Chen *Sampling signals beyond Nyquist frequency Hence discretetime Fourier transform is a frequency scaled version of the continuoustime Fourier transform: 7 X ( e jω) = X s (jω) Ω=ω/Ts Using (16), we finally have X ( e jω) = 1 X c (j ( Ω 2π k)) Ω= ω Ts (18) 9.4 *Sampling signals beyond Nyquist frequency Recall that: Sampling maps the continuoustime frequency π < Ω < π onto the unit circle Sampling also maps the continuoustime frequencies π < Ω < 3 π, 3 π < Ω < 5 π, etc, onto the unit circle Consider sampling signals beyond Nyquist frequency: From (18), sampling maps frequency components beyond Nyquist frequency onto the same discretetime frequency region [ π, π]. The mapping is periodic: portions of the continuoustime jω axis, for Ω in the ranges of ] [ πts, πts, [ [ π, 3π 3π ], and, 5π ], etc, map to the same unit circle in the discretetime domain. 7 Understanding the scaling: in time domain, consecutive samples in x s (t) are spaced by the sampling time ; samples in x [n] are indexed by integers and hence consecutive samples are spaced by 1. The normalization by 1/ in time domain corresponds to a normalization by in the frequency domain, hence ω = Ω. 47 version September 23, 2015

7 X. Chen *Sampling signals beyond Nyquist frequency I π π T u = 2 π 1.2π = 0.6π T u R e j0.6π I 1 1 R (e j1.2π ) e j0.8π I R π 1.2π = 0.6π T u e j0.6π (e j1.2π ) e j0.8π π T u = 2 π s plane Sampled at T u z plane Sampled at z plane Figure 7 Example: sampling signals beyond Nyquist frequency For instance, consider a continuoustime sinusoidal signal cos (Ω o t) with frequency satisfying Ω o = 1.2π/ = 1.2π rad/sec. The continuoustime Fourier transform of the signal is πδ (Ω 1.2π) + πδ (Ω + 1.2π) (The signal contains a positive and a negative frequency components at the same frequency). If sampled at T u = /2, Ω o is below the Nyquist frequency π/t u = 2π rad/sec, and mapped to ω o = Ω o T u = 0.6π. The Fourier transform of the sampled signal is [by using (18)] 2 π [δ (ω 0.6π + 2πk) + δ (ω + 0.6π + 2πk)]. If sampled at, Ω o is beyond Nyquist frequency. Aliasing occurs and the discretetime Fourier transform of the sampled signal becomes πδ (ω ± 1.2π + 2πk) = πδ (ω ± 0.8π + 2πk), i.e., within Nyquist frequency, the observable spectral peaks are at 0.8π/ = 0.8π rad/sec and 0.8π/ = 0.8π rad/sec, instead of the actual frequency 1.2π rad/sec. Graphically, the result is demonstrated in Fig version September 23, 2015