diff --git a/src/discovery/signals.py b/src/discovery/signals.py
index 9a23bc9..68bd006 100644
--- a/src/discovery/signals.py
+++ b/src/discovery/signals.py
@@ -162,19 +162,36 @@ def makegp_ecorr(psr, noisedict={}, enterprise=False, scale=1.0, selection=selec
     backend_flags = selection(psr)
     backends = [b for b in sorted(set(backend_flags)) if b != '']
     masks = [np.array(backend_flags == backend) for backend in backends]
+
+    if len(np.unique(masks)) == 1:
+        # for those pulsar with only one backend
+        first_valid_bin = 0
+    else:
+        first_valid_bin = 1
+
     for backend, mask in zip(backends, masks):
         log10_ecorrs.append(f'{psr.name}_{backend}_log10_ecorr')
 
-        # TOAs that do not belong to this mask get index zero, which is ignored below.
-        # This will fail if there's only one mask that covers all TOAs
         bins = quantize(psr.toas * mask)
 
         if enterprise:
             # legacy accounting of degrees of freedom
-            uniques, counts = np.unique(quantize(psr.toas * mask), return_counts=True)
-            Umats.append(np.vstack([bins == i for i, cnt in zip(uniques[1:], counts[1:]) if cnt > 1]).T)
+            uniques, counts = np.unique(bins, return_counts=True)
+            epoch_masks = [bins == i for i, cnt in zip(
+                uniques[first_valid_bin:],
+                counts[first_valid_bin:]) if cnt > 1]
+
+            if epoch_masks:
+                U_backend = np.vstack(epoch_masks).T
+            else:
+                # if there is no ToAs observed at the same time
+                U_backend = np.zeros((len(bins), 0))
+
+            Umats.append(U_backend)
         else:
-            Umats.append(np.vstack([bins == i for i in range(1, bins.max() + 1)]).T)
+            Umats.append(np.vstack([bins == i for i in range(
+                first_valid_bin, bins.max() + 1)]).T)
+
     Umatall = np.hstack(Umats)
     params = log10_ecorrs