Algorithm 2. Signal modulation classification algorithm based on improved stacking.
Define: (1) The dataset consists of a training set T and a validation set V with sample sizes t and v respectively; (2) The features proposed in Section 2.1 are defined as base-features F with the dimension f, and the class labels of the signal samples are taken as the expected outputs; (3) The number of folds for cross-validation is denoted as K, and the corresponding initialization index k is set to 1. The number of base-learner types is denoted as N, and the corresponding initialization index n is set to 1.
Input: Base-features
Output: Modulation type of signal
Training Phase
Training of base-learners:
1	Divide the training set T into K parts equally; the k-th part serves as the validation set Tak for the base-learner, containing $\frac{t}{K}$ samples; the remaining k−1 parts serve as the training set Tbk for the base-learner, containing $\frac{(K - 1)t}{K}$ samples
2	while (k < K) do
3	while (n < N) do
4	Initialize base-learner $B_n$
5	$B_{(n, k)}=train(B_n, T{b}_k)$ // Train base-learner $B_n$ on training set $T{b}_k$ to obtain the corresponding pre-trained base-learner $B_{(n, k)}$
6	$Q_{(n, k)}=B_{(n, k)}\left(F_k\right)$ // Input base-features $F_k$ to pre-trained base-learner $B_{(n, k)}$ to obtain the corresponding output $Q_{(n, k)}$
7	end while
8	$Q_k=[Q_{(1, k)},Q_{(2, k)},Q_{(3, k)},\cdots ,Q_{(N, k)}]$ // Concatenate $[Q_{(1, k)},Q_{(2, k)},Q_{(3, k)},\cdots ,Q_{(N, k)}]$ along the feature axis to obtain the k-th fold feature $Q_k$
9	Meta-feature enhancement: $P_k=[Q_k,F_k]$ // Concatenate $Q_k$ and $F_k$ along the feature axis to obtain the enhanced meta-feature $P_k$
10	end while
11	$P=[P_1,P_2,P_3,\cdots ,P_K]$ // Concatenate $[P_1,P_2,P_3,\cdots ,P_K]$ along the sample axis to obtain the meta-feature set P, which is based on training set T
12	Save all pre-trained base-learners $B_{(n, k)}$
Training of meta-learner:
13	Initialize the meta-learner $M$
14	$M=train(M, P, O_T)$ // Take the meta-feature set P as the input of meta-learner, take the labels of training set T, i.e., ground truth, as the target output $O_T$ for meta-learner, and then obtain pre-trained meta-learner by training
15	Save the pre-trained meta-learner
Validation Phase
1	Load all pre-trained base-learner $B_{(n, k)}$ and pre-trained meta-learner $M$
2	while (n < N) do
3	while (k < K) do
4	$Q_{(n,k)}=B_{(n,k)}\left(F_V\right)$ // Input the base-feature $F_V$ of validation set V into all pre-trained base-learners $B_{(n,k)}$, and then obtain output $Q_{(n,k)}$
5	end while
6	$Q_n={Voting}_k(Q_{(n,1)}, Q_{(n,2)}, Q_{(n,3)}, \cdots , Q_{(n,K)})$ // Apply the voting method to $Q_{(n,k)}$ to obtain $Q_n$
7	end while
8	Meta-feature enhancement: $P_V=[Q_1,Q_2,Q_3,\cdots ,Q_n,F_V]$ // Concatenate $[Q_1,Q_2,Q_3,\cdots ,Q_n]$ and $F_V$ along the feature axis to obtain enhanced meta-feature $P_V$
9	${R=M(P}_V)$ // Input $P_V$ into pre-trained meta-learner M to obtain prediction R
10	$E=eval(R,O_V)$ // Evaluate the prediction performance of R by comparing with the target output $O_V$ of the validation set V using relevant metrics
11	Return the prediction R and evaluation results E