# Sensing-Assisted Sparse Channel Recovery for Massive Antenna Systems

Zixiang Ren, Graduate Student Member, IEEE, Ling Qiu, Member, IEEE, Jie Xu,
Senior Member, IEEE, and Derrick Wing Kwan Ng, Fellow, IEEE Z. Ren is with Key
Laboratory of Wireless-Optical Communications, Chinese Academy of Sciences,
School of Information Science and Technology, University of Science and
Technology of China, Hefei 230027, China, and the Future Network of
Intelligence Institute (FNii), The Chinese University of Hong Kong (Shenzhen),
Shenzhen 518172, China (e-mail: rzx66@mail.ustc.edu.cn).L. Qiu is with Key
Laboratory of Wireless-Optical Communications, Chinese Academy of Sciences,
School of Information Science and Technology, University of Science and
Technology of China, Hefei 230027, China (e-mail: lqiu@ustc.edu.cn).J. Xu is
with the School of Science and Engineering (SSE) and the FNii, The Chinese
University of Hong Kong (Shenzhen), Shenzhen 518172, China (e-mail:
xujie@cuhk.edu.cn).D. W. K. Ng is with the University of New South Wales,
Sydney, NSW 2052, Australia (e-mail: w.k.ng@unsw.edu.au).L. Qiu and J. Xu are
the corresponding authors.

###### Abstract

This correspondence presents a novel sensing-assisted sparse channel recovery
approach for massive antenna wireless communication systems. We focus on a
fundamental configuration with one massive-antenna base station (BS) and one
single-antenna communication user (CU). The wireless channel exhibits sparsity
and consists of multiple paths associated with scatterers detectable via radar
sensing. Under this setup, the BS first sends downlink pilots to the CU and
concurrently receives the echo pilot signals for sensing the surrounding
scatterers. Subsequently, the CU sends feedback information on its received
pilot signal to the BS. Accordingly, the BS determines the sparse basis based
on the sensed scatterers and proceeds to recover the wireless channel,
exploiting the feedback information based on advanced compressive sensing (CS)
algorithms. Numerical results show that the proposed sensing-assisted approach
significantly increases the overall achievable rate than the conventional
design relying on a discrete Fourier transform (DFT)-based sparse basis
without sensing, thanks to the reduced training overhead and enhanced recovery
accuracy with limited feedback.

###### Index Terms:

Massive antenna system, sparse channel recovery, integrated sensing and
communications (ISAC), compressive sensing (CS).

##  I Introduction

Deploying massive antennas at base stations (BSs) has attracted a lot of
attention in beyond fifth-generation (B5G) and sixth-generation (6G) wireless
networks. Such massive antenna systems provide significantly increased spatial
multiplexing, beamforming, and diversity gains, as well as channel hardening
effects, thus enhancing data-rate throughput, lowering transmission latency,
and improving communication reliability. To fully reap these benefits, it is
imperative for the massive-antenna BS to acquire accurate channel state
information (CSI). This, however, presents practical challenges, especially
for downlink systems. For instance, conventional massive antenna systems
employ pilot-based channel estimation relying on the minimum mean squared
error (MMSE) principle, which induces significant pilot overheads
corresponding to the substantial quantity of transmit antennas [1]. To
overcome this challenge, various prior works (see, e.g., [2]) have advocated
reducing the pilot overheads and enhancing the communication performance by
utilizing the inherent sparsity of massive antenna channels resulting from the
limited scatterers in the environment, especially in high frequency bands such
as millimeter wave (mmWave) and terahertz (THz).

Sparse channel estimation is implemented based on compressed sensing (CS)
techniques [3, 4, 5, 6]. In this paradigm, the BS first transmits a limited
number of pilots (i.e., fewer than the large number of antennas).
Subsequently, after receiving the pilot signals, the CU sends back processed
pilot information to the BS. By capitalizing on the sparse nature of massive-
antenna channels and based on the limited feedback, the BS can recover the
wireless channel via well-established CS algorithms. For instance, the authors
in [3, 4] presented basic pursuit (BP) based CS methods for sparse channel
estimation, in which the discrete Fourier transform (DFT) matrix is exploited
as the sparse basis for representing the channel. Furthermore, the authors in
[5] proposed a dictionary learning approach to dynamically select a sparse
basis from an overcomplete DFT matrix. Nonetheless, this method suffers from
the high computational complexity of the overcomplete DFT matrix and the
associated overhead of dictionary learning. In addition, recent work [6]
studied the representation and estimation of sparse channels in the near-field
by considering the sparsity in both distance and angular domains. However,
these prior designs may suffer from compromised performance and/or enhanced
computational complexity due to the heuristically chosen sparse basis (e.g.,
the over-complete DFT matrix) and the additional cost of dictionary learning.
Therefore, selecting an appropriate sparse basis for concise sparse channel
representation remains an essential yet challenging task.

Recently, integrated sensing and communications (ISAC) has emerged as a
crucial technology for 6G wireless networks, where radar sensing is integrated
into wireless communications to enhance resource utilization efficiency and
foster mutual benefits [7]. Among various ISAC design paradigms, exploiting
environmental sensing to assist channel estimation and wireless communications
is particularly appealing. For example, the authors in [8] proposed a strategy
where the BS sends downlink pilots and conducts target sensing, while the CU
transmits uplink pilots. This strategy enables the BS to estimate the downlink
communication channel by jointly exploiting downlink sensing results and
received uplink pilots. Meanwhile, the authors in [9] explored a scenario
involving practical codebook feedback. Here, the BS transmits downlink pilots
and performs target sensing, while the CU estimates the downlink channel and
subsequently provides practical codebook feedback to the BS. Furthermore, the
authors in [10] jointly investigated the target detection and channel
estimation problem via the common sparsity of communication and sensing
scatterers by jointly utilizing both uplink and downlink pilots. Nevertheless,
[8, 9, 10] share a common challenge that the BS needs to transmit a
substantial number of pilots (exceeding the antenna count). By combining radar
sensing and sparse channel estimation, we can precisely identify a proper
sparse basis for CS signal recovery, thus motivating our work.

This correspondence proposes leveraging ISAC for efficient sparse channel
estimation in massive antenna systems with radar sensing. We focus on a
fundamental configuration featuring one massive-antenna BS and one single-
antenna CU. Within this framework, the BS simultaneously transmits downlink
pilots to the CU while receiving echo signals for scatterer sensing, and then
the CU provides feedback on its received pilots to the BS. Leveraging this
feedback, the BS identifies a sparse basis and employs CS algorithms to
accomplish channel recovery. Our numerical results confirm the superiority of
the sensing-assisted approach over conventional designs relying on a DFT-based
sparse basis without sensing in terms of the overall achievable rate, thanks
to the reduced training overhead and improved accuracy with limited feedback.

Notations: We use boldface lower- and upper-case letters to denote vectors and
matrices, respectively. The space of $N\times M$ complex matrices is
represented by $\mathbb{C}^{N\times M}$. $\boldsymbol{I}$ stands for an
identity matrix, while $\boldsymbol{0}$ represents an all-zero matrix with
appropriate dimensions. For a complex arbitrary-size matrix $\boldsymbol{B}$,
we use $\textrm{rank}(\boldsymbol{B})$, $\boldsymbol{B}^{T}$,
$\boldsymbol{B}^{H}$, and $\boldsymbol{B}^{c}$ to denote its rank, transpose,
conjugate transpose, and complex conjugate, respectively.
$\mathcal{CN}(\boldsymbol{x},\boldsymbol{Y})$ denotes a circularly symmetric
complex Gaussian (CSCG) random vector with mean vector $\boldsymbol{x}$ and
covariance matrix $\boldsymbol{Y}$. The Euclidean norm of a vector is
represented by $\|\cdot\|$. $\|\cdot\|_{0}$ denotes the zero-norm of a vector.
$\mathcal{U}(\cdot)$ denotes a uniformly distributed random variable.
$\boldsymbol{A}\otimes\boldsymbol{B}$ represents the Kronecker product of two
matrices $\boldsymbol{A}$ and $\boldsymbol{B}$. $\mathrm{diag}(\cdot)$ denotes
a diagonal matrix with all non-diagonal elements being zeros, and the diagonal
elements determined by the input.

##  II System Model

Figure 1: Illustration of the massive antenna system.

Fig. 1 shows a sensing-assisted massive antenna communication system that
comprises a multi-antenna ISAC BS featuring a uniform planar array (UPA) of
$N_{v}\times N_{h}$ transmit antennas communicating with a single-antenna
CU111Extending this approach to multi-user or multi-antenna CUs remains an
area for future exploration.. Here, $N_{v}$ and $N_{h}$ denote the vertical
and horizontal antenna numbers, respectively. Within the wireless environment,
there are $M$ scatterers, denoted by set $\mathcal{M}=\\{1,\dots,M\\}$. It is
assumed that only a subset of the environmental scatterers, identified by set
$\mathcal{M}_{c}=\\{1,\cdots,M_{c}\\}\subseteq\mathcal{M}$, render a
significant impact on the communication channel, while other paths are blocked
or ignored, in line with earlier studies [8, 9]. As a result, the channel from
the BS to the CU is expressed as[9]

$\boldsymbol{h}=\sum_{m=1}^{M_{c}}\alpha_{m}\boldsymbol{a}(\theta_{m},\varphi_{m})=\sum_{m=1}^{M}\alpha_{m}\boldsymbol{a}(\theta_{m},\varphi_{m}),$
(1)

where $\alpha_{m}\in\mathbb{C}$ denotes the channel coefficient associated
with scatterer $m$, incorporating the signal propagation path loss and the
scatterer’s radar cross section (RCS), with

$\begin{cases}\alpha_{m}\neq 0,&m\in\mathcal{M}_{c},\\\
\alpha_{m}=0,&m\in\mathcal{M},m\notin\mathcal{M}_{c}.\end{cases}$ (2)

Here, $\theta_{m}$ and $\varphi_{m}$ denote the associated elevation and
azimuth angles of departure of path $m$, respectively, and
$\boldsymbol{a}(\cdot)$ denotes the steering vector of the transmit antenna
array, i.e.,

$\displaystyle\boldsymbol{a}_{v}(\theta_{m})=\frac{1}{N_{v}}[1,e^{j2\pi\frac{d_{v}}{\lambda}\sin\theta_{m}},\dots,e^{j2\pi\frac{d_{v}}{\lambda}(N_{v}-1)\sin\theta_{m}}]^{T},$
(3) $\displaystyle\boldsymbol{a}_{h}(\theta_{m},\varphi_{m})$
$\displaystyle=\frac{1}{N_{h}}[1,e^{j2\pi\frac{d_{h}}{\lambda}\cos\theta_{m}\sin\varphi_{m}},\dots,e^{j2\pi\frac{d_{h}}{\lambda}(N_{h}-1)\cos\theta_{m}\sin\varphi_{m}}]^{T},$
$\displaystyle\boldsymbol{a}(\theta_{m},\varphi_{m})=\boldsymbol{a}_{v}(\theta_{m})\otimes\boldsymbol{a}_{h}(\theta_{m},\varphi_{m}).$

$\boldsymbol{a}_{v}(\theta_{m})$ and
$\boldsymbol{a}_{h}(\theta_{m},\varphi_{m})$ represent the steering vectors
related to the elevation and azimuth angular perturbations, respectively,
where $\lambda$ represents the wavelength, while $d_{v}$ and $d_{h}$ represent
the spacing between two vertically and horizontally adjacent antennas,
respectively. The number of scatterers influencing the communication channel
is often limited due to the restricted angle spread [11, 4]. Regarding this
characteristic, researchers have advocated the exploration of sparsity in the
angular domain to reduce the training overhead [4]. In this context, CS is
often regarded as a promising method.

###  II-A Conventional CS-Based Sparse Channel Estimation

In the conventional approach, the BS first transmits downlink pilots to the
CU. Subsequently, the CU provides feedback on the received pilots222We
consider the frequency division duplex (FDD) systems, in which the
conventional channel reciprocity is generally not applicable.. The BS then
proceeds to estimate the channel by exploiting the feedback through CS.
Finally, the BS transmits data based on the estimated channel [4]. Let us
assume that the total coherent block length is $T$ and the length of downlink
pilots is $K$. The total received downlink pilots by the CU are denoted as

$\boldsymbol{y}_{d}=\boldsymbol{X}_{d}\boldsymbol{h}+\boldsymbol{z}_{d},$ (4)

where $\boldsymbol{X}_{d}\in\mathbb{C}^{K\times N_{v}N_{h}}$ represents the
transmitted downlink pilots and $\boldsymbol{z}_{d}\in\mathbb{C}^{K\times 1}$
is the Gaussian noise term, i.e.,
$\boldsymbol{z}_{d}\sim\mathcal{CN}(0,\sigma^{2})$ with $\sigma^{2}$ denoting
the noise power. After receiving $\boldsymbol{y}_{d}$, the CU feeds the
quantized version $\bar{\boldsymbol{y}_{d}}$ back to the BS.

In order to recover the CSI based on $\bar{\boldsymbol{y}_{d}}$, the BS
exploits the sparsity with basis
$\boldsymbol{A}_{d}=\boldsymbol{A}_{v}\otimes\boldsymbol{A}_{h}$, where
$\boldsymbol{A}_{v}$ and $\boldsymbol{A}_{h}$ are standard discrete DFT
matrices with dimensions $N_{v}$ and $N_{h}$, respectively. Accordingly, the
channel $\boldsymbol{h}$ is expressed as

$\boldsymbol{h}=\boldsymbol{A}_{d}\bar{\boldsymbol{\alpha}},$ (5)

where $\bar{\boldsymbol{\alpha}}\in\mathbb{C}^{N_{v}N_{h}\times 1}$ is the
sparse coefficients with sparse basis $\boldsymbol{A}_{d}$. As a result, the
conventional downlink channel estimation problem utilizing CS is formulated as

$\displaystyle\underset{\bar{\boldsymbol{\alpha}}\in\mathbb{C}^{N_{v}N_{h}\times
1}}{\textrm{arg min}}$ $\displaystyle\|\bar{\boldsymbol{\alpha}}\|_{0}$
$\displaystyle\textrm{s.t.
}\|\bar{\boldsymbol{y}_{d}}-\boldsymbol{X}_{d}\boldsymbol{A}_{d}\bar{\boldsymbol{\alpha}}\|\leq\varepsilon,$
(6)

where $\varepsilon$ denotes the recovery tolerance. It should be noted that
the CS signal recovery problem (6) is generally considered to be NP-hard. As
such, various greedy-based algorithms are available to tackle this challenge,
including orthogonal matching pursuit (OMP) and sparsity adaptive matching
pursuit (SAMP) [12]. In this particular scenario where the exact sparsity
level information, denoted as $S$, is unavailable, the SAMP algorithm holds
more appeal [12]. Specifically, the SAMP algorithm comprises an inner loop and
an outer loop. The sparsity is progressively expanded stage by stage in the
outer loop. Within the inner loop, the estimated sparsity from the outer loop
is utilized for the recovery of the signal (the channel in our context). For a
more comprehensive understanding of the SAMP algorithm and its application,
please refer to the detailed explanation provided in [12]. Let
$\bar{\boldsymbol{\alpha}}^{*}$ denote the obtained solution to problem (6).
We then obtain the recovered channel as

$\bar{\boldsymbol{h}}=\boldsymbol{A}_{d}\bar{\boldsymbol{\alpha}}^{*}.$ (7)

Subsequently, we adopt the maximum ratio transmission for downlink data
transmission, where the transmit beamforming vector is set to be
$\frac{\sqrt{P}\bar{\boldsymbol{h}}}{\|\bar{\boldsymbol{h}}\|}$ with $P$ being
the maximum transmit power. Consequently, the overall achievable rate is
calculated as

$R=\frac{T-K}{T}\log_{2}\big{(}1+\frac{P|\bar{\boldsymbol{h}}^{H}\boldsymbol{h}|^{2}}{\|\bar{\boldsymbol{h}}\|^{2}}\big{)}.$
(8)

It is important to note that the effectiveness of sparse channel recovery is
intricately connected to sparse basis $\boldsymbol{A}_{d}$. In particular,
this choice significantly affects the sparsity level of $\boldsymbol{h}$,
which directly impacts the overall recovery performance. This thus motivates
us to determine an effective sparse basis through radar sensing.

Figure 2: Transmission protocol for sensing-assisted sparse channel estimation
and wireless communications.

##  III Sensing-Assisted Sparse Channel Recovery

This section proposes a sensing-assisted sparse channel recovery approach, in
which the BS accomplishes the sparse basis selection by acquiring estimates of
angles $\\{\theta_{m}\\}_{m=1}^{M}$, $\\{\varphi_{m}\\}_{m=1}^{M}$ via radar
sensing. Subsequently, the BS reconstructs the CSI $\boldsymbol{h}$ by
leveraging the complex coefficients $\\{\alpha_{m}\\}_{m=1}^{M}$ obtained
through CS signal recovery. In this approach, we obtain environmental side
information through radar sensing. This enables us to dynamically adjust the
sparse basis, leading to performance improvements compared to conventional
designs.

In particular, we propose a framework for downlink transmission in the massive
antenna system assisted by radar sensing as shown in Fig. 2. The BS initially
conducts light training pilot transmission with a training length of $K$, and
simultaneously receives reflected echoes to estimate
$\\{\theta_{m}\\}_{m=1}^{M}$, $\\{\varphi_{m}\\}_{m=1}^{M}$. Next, the BS
receives the channel feedback from the CU333Notably, considering the
utilization of uplink resources for CU’s feedback, the time duration of
feedback is not represented in Fig. 2.. Consequently, the BS can reconstruct
the channel vector as $\tilde{\boldsymbol{h}}$. The overall achievable rate
can be similarly calculated as (8) by replacing $\bar{\boldsymbol{h}}$ as
$\tilde{\boldsymbol{h}}$. In the sequel, we focus on the sparse channel
recovery assisted by radar sensing.

###  III-A Reconstruction of Sparse Basis via Radar Sensing

In this subsection, we consider the radar sensing for sparse basis
reconstruction. To begin with, we focus on the downlink training pilots
transmission. Let
$\boldsymbol{X}_{p}=[\boldsymbol{x}_{p}(1),\boldsymbol{x}_{p}(2),\dots,\boldsymbol{x}_{p}(K)]$
denote the transmitted pilots signal, where
$\boldsymbol{x}_{p}(t)\in\mathbb{C}^{N_{v}N_{h}\times 1},\forall
t\in[1,\dots,K]$. First, our attention turns to the radar sensing, where the
BS employs a colocated UPA consisting of $N_{v}\times N_{h}$ antennas for
receiving the echos and estimating the directions of $M$ scatterers. As a
result, the received echo signals at the BS in symbol $t$ are given as

$\begin{array}[]{cr}&\boldsymbol{y}(t)=\sum_{m=1}^{M}\beta_{m}\boldsymbol{a}(\theta_{m},\varphi_{m})\boldsymbol{a}^{T}(\theta_{m},\varphi_{m})\boldsymbol{x}_{p}(t)+\boldsymbol{\boldsymbol{z}}(t),\\\
&t\in\\{1,\dots,K\\},\end{array}$ (9)

where $\beta_{m}$ denotes the reflection coefficient of the echo channel
associated with scatterer $m$,
$\boldsymbol{\boldsymbol{z}}(t)\in\mathbb{C}^{N_{v}N_{h}\times 1}$ denotes the
received Gaussian noise, i.e.,
$\boldsymbol{\boldsymbol{z}}(t)\sim\mathcal{CN}(\boldsymbol{0},\sigma_{s}^{2}\boldsymbol{I})$
with $\sigma_{s}^{2}$ denoting the noise power. Let
$\boldsymbol{Y}=[\boldsymbol{y}(1),\dots,\boldsymbol{y}(K)]$ denote the total
received echo signals. Consequently, the BS can efficiently estimate
$\\{\theta_{m}\\}_{m=1}^{M},\\{\varphi_{m}\\}_{m=1}^{M}$ based on the received
echoes $\boldsymbol{Y}$ via different spatial signal classification
algorithms, such as multiple signal classification (MUSIC) and estimation of
signal parameters via rotational invariance techniques (ESPRIT). Let
$\\{\hat{\theta}_{m}\\}_{m=1}^{M}$ and $\\{\hat{\varphi}_{m}\\}_{m=1}^{M}$
denote the estimates of $\\{\theta_{m}\\}_{m=1}^{M}$ and
$\\{\varphi_{m}\\}_{m=1}^{M}$, respectively. Here, when the number of
scatterers is significantly smaller than that of transmit antennas, i.e.,
$M\ll N_{v}\times N_{h}$, wireless channels exhibit sparsity in the angular
domain [4].

Next, we identify the sparse basis from the estimated angles
$\\{\hat{\theta}_{m}\\}_{m=1}^{M}$ and $\\{\hat{\varphi}_{m}\\}_{m=1}^{M}$.
Let
$\hat{\boldsymbol{A}}=[\boldsymbol{a}(\hat{\theta}_{1},\hat{\varphi}_{1}),\boldsymbol{a}(\hat{\theta}_{2},\hat{\varphi}_{2}),\dots,\boldsymbol{a}(\hat{\theta}_{M},\hat{\varphi}_{M})]$
and $J=\mathrm{rank}(\hat{\boldsymbol{A}})\leq M$. Suppose that the singular
value decomposition (SVD) of $\hat{\boldsymbol{A}}$ is given by

$\hat{\boldsymbol{A}}=\boldsymbol{U}\boldsymbol{\varSigma}\boldsymbol{V}^{H},$
(10)

where $\boldsymbol{U}\in\mathbb{C}^{N_{v}N_{h}\times N_{v}N_{h}}$ and
$\boldsymbol{V}\in\mathbb{C}^{M\times M}$ are unitary matrices, and
$\boldsymbol{\varSigma}=\left[\begin{array}[]{cc}\boldsymbol{\varSigma}_{1}&\boldsymbol{0}\\\
\boldsymbol{0}&\boldsymbol{0}\end{array}\right]\in\mathbb{C}^{N_{v}N_{h}\times
M}$ with
$\boldsymbol{\varSigma}_{1}=\mathrm{diag}(\lambda_{1},\dots,\lambda_{J})$ and
$\lambda_{1}\geq\dots\geq\lambda_{J}>0$. Moreover, we represent the estimated
channel $\tilde{\boldsymbol{h}}$ sparsely using an orthogonal basis as

$\tilde{\boldsymbol{h}}=\hat{\boldsymbol{A}}\boldsymbol{\alpha}=\boldsymbol{U}\boldsymbol{\varSigma}\boldsymbol{V}^{H}\boldsymbol{\alpha}=\boldsymbol{U}\tilde{\boldsymbol{\alpha}},$
(11)

where $\boldsymbol{\alpha}\in\mathbb{C}^{M\times 1}$ is the original path
scattering coefficients vector and
$\tilde{\boldsymbol{\alpha}}=\boldsymbol{\varSigma}\boldsymbol{V}^{H}\boldsymbol{\alpha}$
represents the sparse vector to be recovered. Recall that
$\boldsymbol{\varSigma}=\left[\begin{array}[]{cc}\boldsymbol{\varSigma}_{1}&\boldsymbol{0}\\\
\boldsymbol{0}&\boldsymbol{0}\end{array}\right]\in\mathbb{C}^{N_{v}N_{h}\times
M}$ and $\boldsymbol{V}^{H}\boldsymbol{\alpha}$ is an $M\times 1$ vector,
there can be a maximum of $J$ non-zero elements within
$\tilde{\boldsymbol{\alpha}}$. As a result, we can adopt $\boldsymbol{U}$ as
the sparse basis for sparse signal recovery of $\tilde{\boldsymbol{\alpha}}$.

###  III-B Sparse Channel Recovery based on Feedback

Then, the received signal at the CU is expressed as

$\boldsymbol{y}_{p}=\boldsymbol{X}_{p}\boldsymbol{h}+\boldsymbol{z},$ (12)

where $\boldsymbol{z}\in\mathbb{C}^{K\times 1}$ denotes the Gaussian noise at
the CU receiver, i.e.,
$\boldsymbol{\boldsymbol{z}}\sim\mathcal{CN}(\boldsymbol{0},\sigma_{c}^{2}\boldsymbol{I})$,
where $\sigma_{c}^{2}$ is the noise power. The CU needs to extract essential
information from the received signal $\boldsymbol{y}_{p}$ and feed it back to
the BS. Supposing that the CU feeds back $B$ bits of information, the feedback
signal is expressed as

$\boldsymbol{q}=\mathscr{F}(\boldsymbol{y}_{p}),$ (13)

where function $\mathscr{F}(\cdot):\mathbb{C}^{K\times 1}\rightarrow\\{\pm
1\\}^{B}$ represents the adopted feedback scheme [11]. In particular, in this
work, we employ a random vector quantization (RVQ) codebook for the feedback
of the received vector signal $\boldsymbol{y}_{p}$. In this scheme, the CU
first normalizes the vector $\boldsymbol{y}_{p}$ as
$\bar{\boldsymbol{y}_{p}}=\frac{\boldsymbol{y}_{p}}{\|\boldsymbol{y}_{p}\|}$,
and then feeds back the codeword $\hat{b}$ satisfying

$\hat{b}=\underset{b\in\\{1,2,\cdot\cdot\cdot,2^{B}\\}}{\textrm{arg
max}}|\bar{\boldsymbol{y}_{p}}^{H}\boldsymbol{c}_{b}|^{2},$ (14)

where
$\boldsymbol{C}=[\boldsymbol{c}_{1},\boldsymbol{c}_{2},\dots,\boldsymbol{c}_{2^{B}}]\in\mathbb{C}^{K\times
2^{B}}$ is the pre-defined $B$-bit RVQ codebook444RVQ has been widely adopted
due to its ease of codebook construction and suitability for low-rate feedback
[13]. It is worth noting that other codebook methods, such as Grassmannian
Manifolds or DFT-based approaches, are also applicable.. Assume that the BS
can perfectly obtain the codeword feedback $\hat{b}$ and
$\hat{\boldsymbol{y}_{p}}$ denotes the vector in the codebook mapped by the
codeword $\hat{b}$.

Based on the feedback $\hat{\boldsymbol{y}_{p}}$ together with the sparse
basis constructed via radar sensing in (11), we formulate the CS signal
recovery problem as

$\arg\underset{\tilde{\boldsymbol{\alpha}}}{\min}\|\tilde{\boldsymbol{\alpha}}\|_{0},\mathrm{\textrm{
s.t.
}}\|\hat{\boldsymbol{y}_{p}}-\boldsymbol{X}_{p}\boldsymbol{U}\tilde{\boldsymbol{\alpha}}\|\leq\varepsilon.$
(15)

By incorporating the sparsity basis $\boldsymbol{U}$, received feedback
$\hat{\boldsymbol{y}_{p}}$, and via applying the CS-based SAMP algorithm, we
can achieve accurate and effective reconstruction of the sparse signal
$\tilde{\boldsymbol{\alpha}}$. Let $\tilde{\boldsymbol{\alpha}}^{*}$ denote
the obtained solution to problem (15). Consequently, the channel vector
$\tilde{\boldsymbol{h}}$ is efficiently constructed via (11) as

$\tilde{\boldsymbol{h}}=\boldsymbol{U}\tilde{\boldsymbol{\alpha}}^{*}.$ (16)

##  IV Numerical Results

In this section, we illustrate the performance of our proposed sensing-
assisted CSI recovery algorithm. We evaluate the effectiveness of our proposed
sensing-assisted recovery method by comparing it with the conventional
benchmark that relies on a DFT-based sparse basis [12]. We assume that the BS
transmits at a constant power level and the pilot length is set equally in
both the proposed design and the benchmark for a fair comparison. For both the
benchmark and our proposed sparse basis selection designs, we consider two
scenarios with finite feedback and perfect feedback, respectively.

  * •

Finite feedback: The CU feeds back the received signal $\boldsymbol{y}_{p}$
with a finite number of bits.

  * •

Perfect feedback: The CU feeds back the received signal $\boldsymbol{y}_{p}$
with an infinite number of bits, i.e., the feedback of $\boldsymbol{y}_{p}$ is
perfect.

We evaluate the performance of our proposed sensing-assisted sparse basis
selection design with finite feedback and compare it with the following
schemes:

  * •

Benchmark with finite feedback

  * •

Benchmark with perfect feedback

  * •

Upper bound with perfect CSI

  * •

Proposed design with perfect feedback.

In this context, we examine a massive antenna system where a BS is equipped
with a half-wavelength UPA antenna configuration with $N_{v}=N_{h}=8$. The BS
is located at $[0\textrm{ m, }0\textrm{ m, }10\textrm{ m}]$ in an environment
with $M=6$ paths, similar as [11, 9], among which $M_{c}=4$ scatterers
contribute to the communication channel. We assume that the small-scale
complex path gain of each path follows a standard Gaussian distribution, and
the distance $d_{m}$ between scatterer $m$ and the BS is uniformly distributed
in $[80\textrm{ m,}120\textrm{ m}]$. We model $\theta_{m}$ and $\varphi_{m}$
as uniform distributed random variables, i.e.,
$\theta_{m}\sim\mathcal{U}(-5{{}^{\circ}},+5{{}^{\circ}})$ and
$\varphi_{m}\sim\mathcal{U}(-60{{}^{\circ}},+60{{}^{\circ}})$,
$m\in\mathcal{M}$, similar as [11]. First, for the sensing model, the complex
sensing path coefficient $\beta_{m}$ is calculated by
$|\beta_{m}|=\sqrt{\rho_{0}d_{m}^{-2}\times\gamma_{m}d_{m}^{-2}}$, where
$\rho_{0}$ is the reference path loss at distance $1\textrm{ m}$ and is set as
$-40\textrm{ dB}$, and $\gamma_{m}$ is a Gaussian distributed reflection
coefficient associated with the RCS. The phase of $\beta_{m}$ is randomly
sampled from $[-\pi,\pi]$. Then, as for the communication model, we assume
that the scatterer $m=1$ is the desired CU and the complex multipath gain
$\alpha_{m}$ is calculated by
$|\alpha_{m}|=\sqrt{\rho_{0}d_{m}^{-2}\times\delta_{m}r_{m}^{-2}}$, where
$r_{m}$ is the distance between scatterer $m$ and the CU, while $\delta_{m}$
is a Gaussian distributed reflection coefficient. We consider a coherent block
consisting of $T=200$ symbols, and the first $K=16\ll N_{v}N_{h}$ symbols are
adopted for pilots transmission, unless further specified. We perform 1000
random channel realizations for each figure to evaluate the average
performance.

Figure 3: Average achievable rate versus different levels of receive SNR,
12-bit RVQ feedback.

Figure 4: Achievable achievable rate versus different feedback bits of RVQ,
$\mathrm{SNR}=15\textrm{ dB}.$

Fig. 4 shows the average achievable rate versus the received signal-to-noise
ratios (SNR), represented as
$10\log_{10}(\frac{P\|\boldsymbol{h}\|^{2}}{\sigma^{2}})$. In the case of
finite feedback, we consider the use of RVQ with 12 bits. It is observed that
our proposed sensing-assisted sparse basis consistently outperforms the
conventional DFT based basis across all three training and feedback scenarios.
However, there is a performance loss compared with the upper bound primarily
due to the finite feedback in both recovery algorithms. Furthermore, the
finite feedback significantly degrades the performance of the conventional
DFT-based basis approach due to the lower sparsity level of the feedback
vector.

Fig. 4 shows the achievable rate versus the different number of feedback bits
for RVQ. It is observed that our proposed sensing-assisted scheme achieves
satisfactory performance even with limited feedback bits. This is attributed
to the fact that we only need to recover the signal within a small subspace,
enabling a favorite sparsity level and consequently leading to improved
recovery performance. In contrast, the conventional DFT based sparse basis
exhibits poor sparsity, resulting in a substantial decrease in recovery
performance when utilizing finite feedback. Consequently, the conventional
scheme requires a larger number of feedback bits for signal recovery compared
to our proposed sensing-assisted scheme.

Fig. 6 shows the average achievable rate versus the pilot length $K$. It is
observed that the achievable rate initially rises and subsequently declines
with an increasing pilot lengths $K$. This is due to the fact that a higher
number of pilots can lead to a more accurate channel estimation, but can also
reduce the block length available for information transmission that outweighs
the benefits.

Finally, Fig. 6 shows the average achievable rate versus the total coherent
block length $T$. It is observed that the achievable rate initially rises as
$T$ increases and then becomes saturated. This happens because the influence
of the fixed pilot length becomes negligible when $T$ is sufficiently large.

Figure 5: Average achievable rate versus different pilot lengths $K$,
$\mathrm{SNR}=15\textrm{ dB}$.

Figure 6: Average achievable rate versus the total coherent block length $T$,
$\mathrm{SNR}=15\textrm{ dB}$.

##  V Conclusion

This correspondence presented an innovative approach for sparse channel
recovery in massive antenna wireless communication systems, leveraging radar
sensing. Our method integrated the transmission of downlink pilots with
scatterer sensing, user feedback reception, and the utilization of echo
sensing signals for CSI reconstruction via CS-based algorithms. Numerical
results highlighted substantial performance enhancements, including a notable
reduction in training overhead and a diminished dependence on user feedback
when compared to conventional methods that solely rely on a DFT-based sparse
basis. An interesting direction for future research lies in extending the
application of sensing-assisted sparse channel recovery to distributed or
multi-user scenarios, promising to further enhance the versatility and
efficacy of this approach.

## References

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# Kill The Math and Let the Introductory Course Be Born

David Kane

ABSTRACT: Our introductory classes in statistics and data science use too much
mathematics. The key causal effect which our students want our classes to have
is to improve their future performance and opportunities. The more
professional their computing skills (in the context of data analysis), the
greater their likely success. Introductory courses should feature almost no
mathematical/statistical formulas beyond simple algebra.

_KEYWORDS_ : Education, Teaching, Introductory statistics.

## Introduction

In _A Dance with Dragons_ , the fifth book in the _The Game of Thrones_
series, Maester Aemon Targaryen advises Jon Snow, newly elected Lord Commander
of the Night’s Watch, to “Kill the boy and let the man be born.” Maester
Aemon’s point was not that there was anything wrong with a boy’s friendships
and interests. He was not anti-boy. Instead, Aemon recognized that those
attachments prevented Jon Snow from moving into manhood, from devoting his
full energy to what mattered the most for his new role.

This article provides similar advice: _Kill the math and let the introductory
course be born._ The mathematical and statistical formulas found in a typical
introductory course are not evil in and of themselves. Math is fun and
beautiful. Yet it is precisely an over-emphasis on math which has allowed data
science courses to thrive, to meet the informed student demand which
statistics departments have too long ignored. Math is a boyish enthusiasm,
unobjectionable when considered in isolation but inimical to the goals of our
students because it crowds out topics which matter more.

Nolan and Temple Lang (2010) were among the most prominent to recognize this
conflict, insisting that “Computational literacy and programming are as
fundamental to statistical practice and research as mathematics.” In this
article, I will discuss the causal effects which we and our students want our
courses to achieve, connect those effects to some of the specific advice
offered by Nolan and Temple Lang (2010), and recommend replacing large
portions of the mathematics we currently inflict on our students in
introductory courses with lessons in statistical programming. Kaplan (2020)
bemoans “the crawling pace of the integration of modern computing into the
statistics education curriculum.” Mathematics “grips with a dead hand on the
undergraduate [statistics] curriculum,” according to Cobb (2015). Let us break
loose.

## Goals

Which causal effects do we seek to achieve in our classes? Rumsey (2002),
echoing Cobb (1992) and others, argues for “statistical literacy.” Hardin et
al. (2015) want to prepare students to “think with data.” (See also Horton and
Hardin (2015).) Aliaga et al. (2005) used similar terminology, urging us to
“emphasize statistical literacy and develop statistical thinking” among our
students. No reader will disagree with these worthy goals. Or, perhaps a
better word than _goals_ would be _platitudes_. We need to be much more
precise if we want to make informed choices about how to balance math and
programming in our courses.

De Veaux et al. (2017) discuss guidelines for undergraduate data science
programs. None are objectionable. Yet, in aggregate, they fail to provide
clear metrics for making specific trade-offs. This issue applies across our
curriculum but, for purposes of this article, I will focus on introductory
courses. The (vast?) majority of our students only take one or two classes in
statistics. Consider a collection of causal effects which an introductory
course might have. Rubin (1974) defines a causal effect as the difference
between two potential outcomes. For simplicity, compare the effect of spending
10 extra hours of student time on programming versus spending those same 10
hours on mathematics.

We want our students to do well:

  * •

_In the final project for this class._ Such a causal effect is only defined,
of course, for classes with a final project. For those without, however, we
can consider the causal effect on final project performance if the class were,
counterfactually, to have had a final project.

  * •

_In research projects for other classes._ As Bray et al. (2014) and Mair
(2016) explain, reproducible statistical research is impossible without basic
computer skills. If we want our students to do better work in their other
classes, we must give them the tools to do so.

  * •

_As a summer research intern._ We want out students to perform well when they
work on summer research for us, and for other faculty. As every supervisor
knows, independent work outside the classroom is very different from the
generally well-structured tasks involved in class assignments.

  * •

_When working as a summer intern for non-faculty administrators._ The first
three causal effects are not controversial. Preparing our students for
academic pursuits — class projects, summer research — is at the heart of our
pedagogical responsibilities. But why should we be concerned with how well
students do in working for admissions or in the investment office? Still, we
like our non-faculty colleagues and certainly hope our students do good work
for them.

  * •

_When working as a summer intern or full time employee outside the
university._ “Hold on!” shout my fellow faculty. “When did our introductory
statistics class morph into a seminar at the Office of Career Services?” The
answer: When we decided to take our students’ hopes and dreams seriously.
Students know they won’t be in school forever. Almost all of them will spend
the rest of their lives working for, and being evaluated by, non-professors at
non-academic organizations. Problem sets and exams will be left behind, along
with other childish things. Our students want to do well in the outside world.
Do we want that? None of us are so misanthropic that we want them to do
poorly, of course. Yet any honest appraisal of our syllabi would suggest that
helping them to do well outside of academia is not the highest of our
priorities.

  * •

_In getting hired in any of the above roles._ However well we might prepare
students for summer research positions and full time employment, that
preparation counts for naught if students can’t get those jobs. How many of us
give any thought to increasing our students’ chances of being hired? Again, we
aren’t against student success _per se_ , but very few of us explicitly
consider the causal effects which our courses have on this outcome. On the
margin, 10 hours spent on coding increases the odds of a job offer much more
than 10 hours spent on mathematics, however beautiful the mathematics might
be.

Each professor will weigh the relative importance of these causal effects
differently. Some (many?) will put very little weight on the last two. Your
students, however, will put most (almost all?) their weight on getting a
desirable internship/job/career and doing well at it.

These are not the only outcomes professors (and students) care about. There is
nothing evil about understanding the math behind the sampling distribution of
the mean, the confidence interval for a proportion, a null hypothesis test,
the Central Limit Theorem or the Chi-square statistic. The causal effect of
spending more time on math is to increase student comprehension of these
(important!) theories.

The central reason for the rise of data science courses is that students
recognize, correctly, that taking a typical Data Science 101 course increases
their odds of getting the future they want relative to what those odds would
have been if they had taken Statistics 101. But even those data science
classes don’t do as much as they might to improve the odds.

## An Introductory Course

Even after the improvements in statistics education over the last 20 years,
there is still too much math in the typical introductory course. Consider some
of the statistical formulas in De Veaux et al. (2018), a superb introductory
textbook:

  * •

$\sigma(\hat{p})=\sqrt{\frac{pq}{n}}$, the sampling distribution model for a
proportion.

  * •

$SE(\hat{p})=\sqrt{\frac{\hat{p}\hat{q}}{n}}$, the standard error of an
estimated proportion.

  * •

$\bar{y}\pm
t_{n-1}^{*}\times\frac{\sqrt{\frac{\sum(y-\mu)^{2}}{n}}}{\sqrt{n}}$, the one-
sample $t$-interval for the mean.

  * •

$t_{n-1}=\frac{\bar{y}-\mu_{0}}{\frac{\sqrt{\frac{\sum(y-\mu)^{2}}{n}}}{\sqrt{n}}}$,
the one-sample $t$-test for the mean.

  * •

$SD(\hat{p_{1}}-\hat{p_{2}})=\sqrt{\frac{p_{1}q_{1}}{n_{1}}+\frac{p_{2}q_{2}}{n_{2}}}$,
the sampling distribution model for a difference between proportions.

  * •

$SE(\hat{p_{1}}-\hat{p_{2}})=\sqrt{\frac{\hat{p_{1}}\hat{q_{1}}}{n_{1}}+\frac{\hat{p_{2}}\hat{q_{2}}}{n_{2}}}$,
a two-proportion $z$-interval.

  * •

$z=\frac{(\hat{p_{1}}-\hat{p_{2}})-0}{\sqrt{\frac{\hat{p_{1}}\hat{q_{1}}}{n_{1}}+\frac{\hat{p_{2}}\hat{q_{2}}}{n_{2}}}}$,
a two-proportion $z$-test.

  * •

$t=\frac{(\bar{y_{1}}-\bar{y_{2}})-(\mu_{1}-\mu_{2})}{\sqrt{\frac{s^{2}_{1}}{n_{1}}+\frac{s^{2}_{2}}{n_{2}}}}$,
a sampling distribution model for the difference between two means.

  * •

$t=\frac{(\bar{y_{1}}-\bar{y_{2}})-\Delta_{0}}{\sqrt{\frac{s^{2}_{1}}{n_{1}}+\frac{s^{2}_{2}}{n_{2}}}}$,
a two-sample $t$-test for the difference between means.

  * •

$t_{n-1}=\frac{\bar{d}-\Delta_{0}}{\frac{s_{d}}{\sqrt{n}}}$, the paired
$t$-test.

  * •

$\chi^{2}=\sum\frac{(Obs-Exp)^{2}}{Exp}$, the Chi-square statistic.

  * •

$t=\frac{b_{1}-\beta_{1}}{\frac{s_{e}}{\sqrt{n-1}s_{x}}}$, a sampling
distribution for regression slopes.

  * •

$VIF=\frac{1}{(1-R^{2}_{j})}$, the variance inflation factor.

  * •

$SE(\hat{\mu}_{\nu})=\sqrt{SE^{2}(b_{1})\times(x_{\nu}-\bar{x})^{2}+\frac{s^{2}_{e}}{n}}$,
confidence interval for a predicted mean value.

  * •

$SE(\hat{y}_{\nu})=\sqrt{SE^{2}(b_{1})\times(x_{\nu}-\bar{x})^{2}+\frac{s^{2}_{e}}{n}+s^{2}_{e}}$,
the prediction interval for an individual.

I use De Veaux et al. (2018) for this example, not because it is excessively
mathematical, but because it represents the very best of accepted practice,
including a solid introduction to randomization-based inference. With regard
to the changes made for the current edition, the authors insist (p. xi) that
“all of these enhancements follow the new Guidelines for Assessment and
Instruction in Statistics Education (GAISE) 2016 report” resulting in “a
course that is more aligned with the skills needed in the 21st century, one
that focuses even more on statistical thinking and makes use of technology in
innovative ways, while retaining core principles and topic coverage.” And who
is better positioned than second author Paul Velleman, a co-author of both the
GAISE 2005 and GAISE 2016 reports, to make such a claim?

The central problem with curriculum design, as with writing, is knowing what
to cut. De Veaux et al. (2018) explicitly refuse that hard choice, noting that
“Many first statistics courses serve wide audiences of students who need these
skills for their own work in disciplines where traditional statistical methods
are, well, traditional. So we have not reduced our emphasis on the concepts
and methods you expect to find in our texts.”

In other words, despite decades of committee meetings and reports, these
authors present — and expect their students to learn — about the same amount
of mathematics as they would have last century. I urge a different approach.

Consider Gov 50: Data, a Harvard course which I have taught for the last 4
semesters. (Kane (2020a)) This is an introductory statistics course, focused
on meeting the needs of political science majors but open to the entire
university. In the most recent semester, we had 120 students, ranging in level
from first years through masters students, with the typical student having no
background in statistics or programming. We use an open-source textbook (Kane
(2020b)) created specifically for the course. One motto for the course is
that, even though everyone is welcome, the target audience is “poets and
philosophers,” students whose main academic focus is elsewhere but who
recognize that some basic data analysis skills will be helpful, in both their
other courses and after graduation. The course is designed to be the answer to
the following question: “If I only want to devote one course to learning
skills which are valued outside the university, which course should I take?”

I never discuss the formulas which De Veaux et al. (2018) spend so much time
on. I never ask a question about them in problem sets or exams. I devote all
that time, and more, toward improving students’ computational skills. My
advice:

  * •

_Use Git and Github_. See Bryan (2018), Fiksel et al. (2019) and Beckman et
al. (2020) for guidance. It is possible to write an essay using just a
typewriter. It is possible to write code without using source control. Neither
is professional. Although Git/Github are the most popular tools today, that
will change over time. The important point is not the exact tool. The key is
that students should, as much as possible, work with professional tools in a
professional fashion. No employer wants to teach your students about source
control.

  * •

_Use an open source programming language_. For introductory classes, R (R Core
Team (2019)) is the best choice. See Çetinkaya-Rundel and Bray (2012), Carson
and Basiliko (2016), Silva and Moura (2020) and Long and Turner (2020) and for
discussion. Python can also work, especially if some prior programming
experience is a prerequisite for the class. These are the statistical
programming languages that employers want our students to know.

  * •

_Teach randomization-based inference_. Cobb (2007) provides some background on
why we should prefer such an approach. See Wardrop (1995) for an early
approach and Ernst (2004) for an overview. Ismay and Kim (2019) is an
excellent textbook, suitable for use in an introductory class. Although there
is evidence (Hildreth et al. (2018)) that a simulation-based curriculum
improves student understanding, an even bigger advantage, relative to the
traditional approach, is the excuse it offers for programming practice. The
more code that students write, the better they will become at coding.

  * •

_Flip the classroom_. Nielson et al. (2018) demonstrate that students learn
more in a flipped classroom. Perhaps even more important, however, is that a
flipped classroom allows for more time to be spent in supervised practice.
Every minute spent confronting real data — importing, shaping, visualizing,
modeling — is a minute well-spent. Can the same be said for every minute of
every lecture? If students will only devote X hours per week to an
introductory course, the best way to ensure that they got as much practice as
possible is to devote class time to that practice. You learn soccer with the
ball at your feet. You learn data science with your hands on the keyboard.

  * •

_Cold-call_. There is no better way to ensure that students are engaged in a
class than to call on them at random. See Dallimore et al. (2012) for a
literature review and Lemov (2015) for practical details.

  * •

_Require that student work be reproducible_. See Bray et al. (2014) for
motivation and Baumer et al. (2014) for advice. If your own work is
reproducible (and I assume that it is), why wouldn’t you demand the same level
of rigor from your students?

  * •

_Require that student work be public_. Students are somewhat diligent with the
work that only you will see. They take work which others will see much more
seriously. Requiring that student work be public is the easiest way, or at
least the most pleasant way, to cause students to work harder. And the harder
they work, the more likely they are to learn something. Naive observers might
fear that government regulations, like the Family Educational Rights and
Privacy Act (FERPA), prevent faculty from requiring work be made public. This
is not true. See Ramirez (2009) for details. FERPA, and similar regulations,
apply to the records — grades, comments, et cetera — which faculty create. You
can’t make those public. You may require that student work be public. See
bit.ly/1005_projects for several hundred projects completed by my students.

  * •

_Require solo final projects_. See Ledolter (1995) for background and
discussion of final (or research) projects in statistics classes. Imagine that
you are trying to decide between two otherwise similar candidates for a summer
research position. The first has created an impressive final project, written
in R and hosted on Github, for which all the analysis is reproducible. The
second has an impressive grasp of the mathematics behind a $t$-test. Who would
you hire? Outside of academia, where almost no one uses much mathematics, the
first student has a big advantage, just as Horton (2015) reports. Projects
should be solo because otherwise students will divide-and-conquer the work,
often with one doing all the writing and the other all the coding, which is
not the outcome we want.

My introductory course does all these things. The combined causal effects of
these requirements is to significantly increase my students’ odds of achieving
the futures they want, relative to what those odds would have been if my
course were more mathematical. Computational skills are more relevant than
mathematical understanding to their future success.

A similar course would work just as well outside of Harvard. First, there is
an extensive overlap between the bottom quarter of the Harvard distribution of
student ability and the student bodies of other top tercile schools. Second,
weaker students are even more interested in learning skills which employers
value. With less talented students, I would follow exactly the same approach
while, perhaps, cutting back on the total number of topics covered.

## Conclusion

Horton (2015) writes:

> Anecdotal reports have indicated that statistics undergraduates are at a
> competitive disadvantage relative to undergraduate computer science (CS)
> degree holders for entry level positions. Such positions tend to have data-
> related skills at the core of the job descriptions. At present, many CS
> students tend to be able to perform computations with data more easily than
> their statistics equivalents. This should not be allowed to continue.

“Allowed by whom?” one might ask. The central problem is that, even a decade
after Nolan and Temple Lang (2010) made a similar point, we have done too
little. Nolan and Temple Lang (2010) quoted Friedman (2001):

> Computing has been one of the most glaring omissions in the set of tools
> that have so far defined Statistics. Had we incorporated computing
> methodology from its inception as a fundamental statistical tool (as opposed
> to simply a convenient way to apply our existing tools) many of the other
> data related fields would not have needed to exist. They would have been
> part of our field.

Indeed. But it is never too late to change course. Do you need to make the
full leap described above next semester? No. Start slowly. Remove 10 hours of
time spent on math and add 10 hours more on programming.

Friedman (2001) also suggested that “We may have to moderate our romance with
mathematics.” There is no “may” about it. To make room for computation, we
must ditch something. The most obvious something is mathematics. We should
stop using mathematics beyond algebra in our introductory courses. No question
on a problem set or exam should require the use of a formula. We need radical
surgery if our courses are to have the causal effects which our students most
want. My message to instructors: Stop using mathematics, despite your fond
memories of how it helped you in the past, and, from this point forward, use
only computers. _Kill the math and let the introductory course be born._

_Acknowledgment_ : I thank Joe Blitzstein, Mike Parzen and Liberty Vittert for
useful discussions. Special thanks to three anonymous reviewers and to as
associate editor of the _Journal of Statistics Education_ for their comments,
and to an anonymous reviewer and an editor at the _Harvard Data Science
Review_.

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