  This paper finds new tight finite-blocklength bounds for the best achievable
lossy joint source-channel code rate, and demonstrates that joint
source-channel code design brings considerable performance advantage over a
separate one in the non-asymptotic regime. A joint source-channel code maps a
block of $k$ source symbols onto a length$-n$ channel codeword, and the
fidelity of reproduction at the receiver end is measured by the probability
$\epsilon$ that the distortion exceeds a given threshold $d$. For memoryless
sources and channels, it is demonstrated that the parameters of the best joint
source-channel code must satisfy $nC - kR(d) \approx \sqrt{nV + k \mathcal
V(d)} Q(\epsilon)$, where $C$ and $V$ are the channel capacity and channel
dispersion, respectively; $R(d)$ and $\mathcal V(d)$ are the source
rate-distortion and rate-dispersion functions; and $Q$ is the standard Gaussian
complementary cdf. Symbol-by-symbol (uncoded) transmission is known to achieve
the Shannon limit when the source and channel satisfy a certain probabilistic
matching condition. In this paper we show that even when this condition is not
satisfied, symbol-by-symbol transmission is, in some cases, the best known
strategy in the non-asymptotic regime.
