Conversion of Continuous Signals and Systems to Discrete Time

BIBLIOGRAPHY

2.11 Conversion of Continuous Signals and Systems to Discrete Time

2.11.1 Sampling Theorem

We began this review by sampling a continuous-time speech waveform to generate a discrete-time sequence, i.e.,x[n]=x_a(t )|t=nT, whereT is the sampling interval. An implicit assumption was that the sampling is fast enough so that we can recoverx_a(t ) _fromx_[n]. The condition under which recovery is possible is called theSampling Theorem.

Sampling Theorem: Suppose thatx_a(t ) is sampled at a rate of F_s = _T¹ samples per second and supposex_a(t ) is a bandlimited signal, i.e., its continuous-time Fourier transformX_a()is such thatX_a()=0 for|| ≥_N =2π F_N. Thenx_a(t )can be uniquely determined from its uniformly spaced samplesx[n]=x_a(nT )if the sampling frequencyF_s is greater than twice the largest frequency of the signal, i.e., F_s >₂F_N. The largest frequency in the signalF_N is called theNyquist frequency, and 2F_{N, which} must be attained in sampling for reconstruction, is called theNyquist rate.

For example, in speech we might assume a 5000 Hz bandwidth. Therefore, for signal recovery we must sample at _T¹ = 10000 samples/s corresponding to aT =100μs sampling interval.

The basis for the Sampling Theorem is that samplingx_a(t )at a rate of_T¹ results in spectral duplicates spaced by _T², so that sampling at the Nyquist rate avoids aliasing, thus preserving the spectral integrity of the signal. The sampling can be performed with a periodic impulse train with spacingT and unity weights, i.e.,p(t ) =_∞

k=−∞δ(t−kT ). The impulse train resulting from multiplication with the signalx_a(t ), denoted byx_p(t ), has weights equal to the signal values evaluated at the sampling rate, i.e.,

x_p(t ) = x_a(t )p(t )

=

∞

k=−∞

x_a(kT )δ(t −kT ). (2.33) The impulse weights are values of the discrete-time signal, i.e.,x[n]=x_a(nT ), and therefore, as illustrated in Figure 2.11, the cascade of sampling with the impulse trainp(t )followed by con- version of the resulting impulse weights to a sequence is thought of as an ideal A/D converter (or C/D converter). In the frequency domain, the impulse trainp(t )maps to another impulse train with spacing 2π F_s, i.e., the Fourier transform ofp(t )isP ()= ²_T^π_∞

k=−∞δ(−k_s) where_s = 2π F_s. Using the continuous-time version of the Windowing Theorem, it fol-

Discrete- Time System

Sequence to Impulse

Train y_p(t) x_p(t)

x_a(t) x[n] y[n]

Sampling

y_a(t) Impulse Train

to Sequence

p(t) =∑δ∞(t–kT) k = –∞

C/D

D/C

×

Ωs

–ΩN ΩN –ΩN ΩN

2 Ωs

2 sin(x)/x Interpolation

2π T

–ΩN

2π T 2π

T

ΩN2π T

–ΩN ΩN Ω Ω ω Ω

ω Ω

–π π

–2π 2π

–π π

–2π 2π

–

–

–

Figure 2.11 Path from sampling to reconstruction (2_N =_s).

lows thatP ()convolves with the Fourier transform of the signalx_a(t ), thus resulting in a continuous-time Fourier transform with spectral duplicates

X_p() = ¹ T

∞

k=−∞

X_a(−k_s) (2.34)

where_s =²π F_s. Therefore, the original continuous-time signalx_a(t )can be recovered by applying a lowpass analog filter, unity in the passband

−₂^s,^s

2

, and zero outside this band.

This perspective also leads to a reconstruction formula which interpolatesthe signal samples with a sin function. Using the continuous-time version of the Convolution Theorem, application of an ideal lowpass filter of width_s corresponds to the convolution of the filter impulse response with the signal-weighted impulse trainx_p(t ). Thus, we have a reconstruction formula given by

x_a(t ) =

∞

n=−∞

x_a(nT )^sin(π(t −nT )/T ) π(t −nT )/T

because the function ^sin_π(t^(π(t₋⁻_{nT )/T}^{nT )/T )} is the inverse Fourier transform of the ideal lowpass filter.

As illustrated in Figure 2.11, the cascade of the conversion of the sequencey[n]=x[n] to a continuous-time impulse train (with weightsx_a(nT )) followed by lowpass filtering is thought of as adiscrete-to-continuous (D/C) converter. In practice, however, adigital-to-analog converter is used. Unlike the D/C converter, because of quantization error and other forms of distortion, D/A converters do not achieve perfect reconstruction.

2.11 Conversion of Continuous Signals and Systems to Discrete Time 45

The relation between the Fourier transform ofx_a(t ),X_a(), and the discrete-time Fourier transform ofx[n]=x_a(nT ),X(ω), can now be deduced. When the Sampling Theorem holds, over the frequency interval [−π, π] X(ω) is a frequency-scaled (or frequency-normalized) version ofX_a(). Specifically, over the interval [−π, π] we have

X(ω) = ¹ TX_a

ω

T

, |ω| ≤ π.

This relation can be obtained by first observing thatX_p()can be written as

X_p() = ^∞

n=−∞

x_a(nT )e^{−j T n} (2.35)

and then by applying the continuous-time Fourier transform to Equation (2.33), and comparing this result with the expression for the discrete-time Fourier transform in Equation (2.2) [7]. We see that if the sampling is performed exactly at the Nyquist rate, then the normalized frequency π corresponds to the highest frequency in the signal. For example, whenF_N = 5000 Hz, then π corresponds to 5000 Hz.

The entire path, including sampling, application of a discrete-time system, and recon- struction, as well as the frequency relation between signals, is illustrated in Figure 2.11. In this illustration, the sampling frequency equals the Nyquist rate, i.e., 2_N =_s. The discrete-time system shown in the figure may be, for example, a digital filter which has been designed in discrete time or derived from sampling an analog filter with some desired properties.

A topic related to the Sampling Theorem is the decrease and increase of the sampling rate, referred to asdecimationandinterpolationor, alternatively, asdownsamplingandupsampling, respectively. Changing the sampling rate can be important in speech processing where one parameter may be deemed to be slowly-varying relative to another; for example, the state of the vocal tract may vary more slowly, and thus have a smaller bandwidth, than the state of the vocal cords. Therefore, different sampling rates may be applied in their estimation, requiring a change in sampling rate in waveform reconstruction. For example, although the speech waveform is sampled at, say, 10000 samples/s, the vocal cord parameters may be sampled at 100 times/s, while the vocal tract parameters are sampled at 50 times/s. The reader should briefly review one of the numerous tutorials on decimation and interpolation [3],[7],[10].

2.11.2 Sampling a System Response

In the previous section, we sampled a continuous-time waveform to obtain discrete-time samples for processing by a digital computer or other discrete-time-based system. We will also have occasion to transform analog systems to discrete-time systems, as, for example, in sampling a continuous-time representation of the vocal tract impulse response, or in the replication of the spectral shape of an analog filter. One approach to this transformation is to simply sample the continuous-time impulse response of the analog system; i.e., we perform the continuous-to- discrete-time mapping

h[n] = h_a(nT )

whereh_a(t )is the analog system impulse response andT is the sampling interval. This method of discrete-time filter design is referred to as theimpulse invariancemethod [7].

Similar to sampling of continuous-time waveforms, the discrete-time Fourier transform of the sequenceh[n],H (ω), is related to the continuous-time Fourier transform ofh_a(t ),H_a(), by the relation

H (ω) = ¹ TH_a

ω

T

, |ω| ≤ π

where we assume h_a(t ) is bandlimited and the sampling rate is such to satisfy the Nyquist criterion [7]. The frequency response of the analog signal is therefore preserved. It is also of interest to determine how poles and zeros are transformed in going from the continuous- to the discrete-time filter domains as, for example, in transforming a continuous-time vocal tract impulse response. To obtain a flavor for this style of conversion, consider the continuous-time rendition of the IIR filter in Equation (2.25), i.e.,

h_a(t ) =

N

k=1

A_ke^sktu(t )

whose Laplace transform is given in partial fraction expansion form (the continuous counterpart to Equation (2.18)) [7]

H_a(s) =

N

k=1

A_k s −s_k.

Then the impulse invariance method results in the discrete-time impulse response

h[n] = h_a[nT] =

N

k=1

A_ke^{(skT )n}u[n] whosez-transform is given by

H (z) =

N

k=1

A_k 1−e^{(skT )}z⁻¹

with poles atz=e^{(skT )}inside the unity circle in thez-plane (|e^{(skT )}| =e^{(Re[sk]T )}<1 when Re[s_k] < 0) is mapped from poles in thes-plane froms = s_k located to the left of thej axis. Poles being to the left of thej axis is a stability condition for causal continuous systems.

Although the poles are mapped inside the unit circle, the mapping of the zeros depends on both the resulting poles and the coefficientsA_k in the partial fraction expansion. It is conceivable, therefore, that a minimum-phase response may be mapped to a mixed-phase response with zeros outside the unit circle, a consideration that can be particularly important in modeling the vocal tract impulse response.