In previous posts we have explored sampling of continuous-time signals and introduced the Z-transform as a tool to work with discrete-time signals. In this post we address the other end of the process, that is the reconstruction of a signal from its samples.<\/p>\n\n\n\n\n\n\n\n
By signal reconstruction we mean the process of generating a continuous-times signal $u_r(t)$ from periodic samples $u[n]$, $n \\in \\mathbb{Z}$, that is the inverse operation of sampling. As with sampling, one can easily construct a physical device, for example a zero-order holder<\/em> to produce the continuous-time signal <\/p>\n\n\n\n $$ from given discrete samples $u[n]$. As with sampling, properly analyzing the behavior of such a device is a much more complicated task.<\/p>\n\n\n\n Note that, in general, if $u[n]$ was obtained by sampling a continuous-time signal $u(t)$, that is $u[n] = u(n T_s)$, that there is no reason to expect that $u_r(t) = u(t)$. Nor that $u_r(t) = u_s(t)$, where $u_s(t)$ is the sample-and-hold version of the signal $u(t)$ as discussed previously<\/a>. Indeed, the possibility of aliasing<\/a><\/em> will generally make it impossible to exactly reconstruct the original continuous-time signals from its samples.<\/p>\n\n\n\n Take for example the family of continuous-time sinusoidal signals<\/p>\n\n\n\n $$ Because<\/p>\n\n\n\n $$ it should not be possible to distinguish between any of the members of the family exclusively from the samples $u[n]$. These signals are aliases<\/em> of each other.<\/p>\n\n\n\n We have two main reasons for studying signal reconstruction: the first is to characterize the continuous-time signal $u_r(t)$ that can be reconstructed from the samples $u[n]$, the second is to derive conditions under which it would be possible to exactly reconstruct a signal from its samples. We address the first in this post.<\/p>\n\n\n\n Recall the Z-transform of a discrete-time signal $u[n]$, $n\\in\\mathbb{N}$, given by<\/p>\n\n\n\n $$U[z]=\\mathcal{Z} \\{ u[n] \\} = \\sum_{n=0}^{\\infty} u[n] \\, z^{-n}.$$<\/p>\n\n\n\n We have shown earlier that the existence of an exponential bound on $u[n]$ is a sufficient condition for $U[z]$ to exist and converge outside of a disk of radius, say, $\\tilde{\\sigma}_u$.<\/p>\n\n\n\n Our first goal is to calculate the signal $u_r(t)$ from $U[z]$. We have shown in this post<\/a> that if $u_t(t)$ is the continuous-time modulated train of impulses obtained from a continuous-time signal $u(t)$ that it is possible to calculate $U_t(s)=\\mathcal{L}\\{u_t(t)\\}$ directly from the corresponding $U[z]$ by evaluating<\/p>\n\n\n\n $$U_t(s)=\\left . U[z] \\right|_{z=e^{s T_s}} = U[e^{s T_s}].$$<\/p>\n\n\n\n Conversely, one would expect that the Laplace inverse formula applied to $U[e^{s T_s}]$, that is<\/p>\n\n\n\n $$ should produce a continuous-time modulated train of impulses. From this train of impulses,$u_z(t)$, a time-invariant filter with impulse response $g_h(t) = 1(t) – 1(t – T_s)$, see this post<\/a> for details, should provide the signal $u_r(t)$!<\/p>\n\n\n\n Moreover, the magnitude of each impulse of the modulated train of impulses, $u_z(t)$, should coincide with the samples used to produce $U[z]$. That is, solving the above signal reconstruction problem actually amounts to inverting the Z-transform!<\/p>\n\n\n\n Start by noticing that <\/p>\n\n\n\n $$U[e^{(s – j k\\omega_s) T_s}]= U[e^{s T_s}], \\quad k \\in \\mathbb{Z},$$<\/p>\n\n\n\n to define<\/p>\n\n\n\n $$ so that<\/p>\n\n\n\n $$U[e^{s T_s}]= \\frac{1}{T_s} \\sum_{k = -\\infty}^{\\infty} \\hat{U}[e^{(s -j k \\omega_s) T_s}].$$<\/p>\n\n\n\n
u_r(t) = u[n], \\quad n T_s \\leq t \\leq (n+1) T_s
$$<\/p>\n\n\n\n
\\begin{aligned}
u_k(t) &= \\cos(\\omega t + k \\omega_s t), &
\\omega_s &= \\frac{2 \\pi}{T_s}, &
k &\\in \\mathbb{Z}.
\\end{aligned}
$$ <\/p>\n\n\n\n
u[n] = u_0(n T_s) = u_1(n T_s) = u_2(n T_s) = \\cdots
$$<\/p>\n\n\n\nSignal reconstruction and the inverse Z-transform<\/h2>\n\n\n\n
u_z(t) =\\mathcal{L}^{-1} \\left \\{ U[e^{s T_s}] \\right \\},
$$ <\/p>\n\n\n\nThe inverse Z-transform<\/h2>\n\n\n\n
\\hat{U}[z] = \\begin{cases}
T_s U[z], & |\\angle z| < \\pi \\\\
0, & |\\angle z| \\geq \\pi
\\end{cases}
$$<\/p>\n\n\n\n