fix(cloudwatch): automatic metric math label cannot be suppressed

packages/@aws-cdk/aws-cloudwatch/lib/metric.ts

@@ -128,6 +128,7 @@ export interface MetricOptions extends CommonMetricOptions {
 export interface MathExpressionOptions {
   /**
    * Label for this metric when added to a Graph in a Dashboard
+   * To have the legend show the original metric labels only, pass an empty string, i.e. label: ''.
    *
    * @default - Expression value is used as label
    */

packages/@aws-cdk/aws-cloudwatch/lib/private/rendering.ts

@@ -66,11 +66,13 @@ function metricGraphJson(metric: IMetric, yAxis?: string, id?: string) {
   if (yAxis !== 'left') { options.yAxis = yAxis; }
   if (id) { options.id = id; }
 
-  // If math expressions don't have a label (or an ID), they'll render with an unelegant
-  // autogenerated id ("metric_alias0"). Our ids may in the future also be autogenerated,
-  // so if an ME doesn't have a label, use its toString() as the label (renders the expression).
   if (options.visible !== false && options.expression && !options.label) {
-    options.label = metric.toString();
+    if (options.label === undefined) {
+      options.label = metric.toString();
+    } else {
+      // if an empty string was passed explicitly, set the label to undefined so CloudWatch will use the original metric legend
+      options.label = undefined;
+    }
   }
 
   const renderedOpts = dropUndefined(options);

packages/@aws-cdk/aws-cloudwatch/test/metric-math.test.ts

@@ -145,6 +145,27 @@ describe('Metric Math', () => {
       ]);
 
 
+    });
+
+    test('passing an empty string as the label of a MathExpressions does not emit a label', () => {
+      const graph = new GraphWidget({
+        left: [
+          new MathExpression({
+            expression: 'a + e',
+            label: '',
+            usingMetrics: {
+              a,
+            },
+          }),
+        ],
+      });
+
+      graphMetricsAre(graph, [
+        [{ expression: 'a + e' }],
+        ['Test', 'ACount', { visible: false, id: 'a' }],
+      ]);
+
+
     });
 
     test('can reuse identifiers in MathExpressions if metrics are the same', () => {