Distortion-Delay Tradeoff for a Gaussian Source
Transmitted over a Fading Channel
Qiang Li and C. N. Georghiades
Electrical and Computer Engineering Department, Texas A&M University
College Station, TX 77843-3128
E-mail: {qiangli,georghiades}@ece.tamu.edu
Abstract— We study the end-to-end distortion-delay tradeoff
for a Gaussian source transmitted over a fading channel. The
analog source is quantized and stored in a buffer until it is
transmitted. There are two extreme cases as far as buffer delay
is concerned: no delay and infinite delay. We observe that there is
a significant power gain by introducing a buffer delay. Our goal is
to investigate the situation between these two extremes. Using the
recently proposed effective capacity concept, we derive a closed-
form expression for this tradeoff. In order to characterize the
convergence behavior, we derive an asymptotically tight upper
bound for our tradeoff curve, which approaches the infinite delay
lower bound polynomially. Numerical results demonstrate that
introduction of a small amount of delay can save significant
transmission power.
I. I NTRODUCTION
Quality-of-Service (QoS) is a critical design objective for
next-generation packet cellular networks. End-to-End distor-
tion and transmission delay are two fundamental QoS metrics.
Such QoS requirements pose a challenge for the system design
due to the unreliability and time varying nature of the wireless
link.
Quantizer Buffer Analog
Source
Adaptive
Transmitter
Ideal CSI feedback
Fading
Channel
Receiver
Fig. 1. System model
In this paper we consider transmission of a Gaussian source
over a wireless time-varying fading channel. Our goal is to
optimize the end-to-end distortion given a delay constraint.
More precisely, we derive the distortion and delay tradeoff for
the wireless fading channel. To this end, we adopt a cross-layer
approach as shown in Figure 1. At this point, for simplicity
we assume an independent and identically distributed (i.i.d.)
block fading channel model. Such a model is suitable for
serval practical communication scenarios, e.g., time hopping
in TDMA, frequency hopping in FDMA and multicarrier
systems. Extension to the more practical time-correlated case
will be discussed later. Throughout this paper, we always
assume channel state information (CSI) is perfectly known
at the transmitter. We consider an i.i.d., real, Gaussian ∼
N (0, 1) source, which is quantized and then fed into a buffer.
Since the channel is time-varying, the transmitter adjusts the
transmission rate to the current channel state. The relevant
performance criteria are end-to-end quadratic distortion and
the buffer delay. We aim to find the relationship between the
distortion and delay for some average transmission power.
For this problem, there are two extreme cases: 1) There is
no buffer, i.e. no delay); 2) we have an infinite buffer size,
i.e., we allow an infinite transmission delay. For the first case,
we adaptively quantize the Gaussian source according to the
CSI. Assuming perfect transmission, we can approximate the
average achievable quadratic distortion by:
D
0
(ρ)= E
exp(−2 log(1 + α
P
N
0
W
)
, (1)
where P denotes the transmission power and W and N
0
the
bandwidth and noise variance; α is the channel gain, a ran-
dom variable with unit variance following a certain statistical
distribution. Here, we have used the Gaussian distortion-rate
function expressed as D(R
s
) = exp(−2R
s
) with C(ρ)=
log(1 + αρ) as the AWGN channel capacity-cost function and
ρ =
P
N0W
. For the infinite delay case, the average transmission
rate can achieve the ergodic capacity of a fading channel and
the quantizer can simply adopt a constant output rate. The
average distortion is given by:
D
∞
(ρ) = exp
−2 E[log(1 + α
P
N
0
W
)]
. (2)
The function exp(−2(·)) is a convex function. By Jensen’s
inequality then, the distortion D
0
is lower-bounded by D
∞
,
i.e., D
0
≥D
∞
. The two distortion functions are plotted in
Figure 2 for a Rayleigh fading channel. Notice that there
is a significant dB gap between the no-delay and infinite
delay curves. We can call this transmission power gap the
“Jensen’s gain”. So introducing a buffer at the transmitter
to match the source rate with the instantaneous quality of
the channel can save significantly in transmission power to
meet some distortion requirement. Also, we have simplified
the quantization step by allowing a constant rate. A natural
question is therefore: if we only allow a finite delay or
buffer, how much gain can we achieve? How fast does the
distortion curve converge to the infinite-delay lower bound as
the delay increases? The main result of this paper is a clear
characterization of the tradeoff between end-to-end quadratic
distortion and delay, which provides insight to the impact of
the buffer delay on the achieved distortion function of the
Gaussian source transmitted over a wireless fading channel.
ISIT 2006, Seattle, USA, July 9 14, 2006
21 1424405041/06/$20.00 ©2006 IEEE