diff --git a/pyfstat/core.py b/pyfstat/core.py
index a5087659a588916ea08773d86ed5485e93f532b6..0c41b5c06c1b6b62e82de8bc02e6b7150c9bed09 100755
--- a/pyfstat/core.py
+++ b/pyfstat/core.py
@@ -330,7 +330,8 @@ class ComputeFstat(BaseSearchClass):
 
     @helper_functions.initializer
     def __init__(self, tref, sftfilepattern=None, minStartTime=None,
-                 maxStartTime=None, binary=False, transient=True, BSGL=False,
+                 maxStartTime=None, binary=False, BSGL=False,
+                 transientWindowType=None, t0Band=None, tauBand=None,
                  detectors=None, minCoverFreq=None, maxCoverFreq=None,
                  injectSources=None, injectSqrtSX=None, assumeSqrtSX=None,
                  SSBprec=None):
@@ -347,8 +348,11 @@ class ComputeFstat(BaseSearchClass):
             this epoch
         binary : bool
             If true, search of binary parameters.
-        transient : bool
-            If true, allow for the Fstat to be computed over a transient range.
+        transientWindowType: str
+            If 'rect' or 'exp',
+            allow for the Fstat to be computed over a transient range.
+            ('none' instead of None explicitly calls the transient-window function,
+             but with the full range, for debugging)
         BSGL : bool
             If true, compute the BSGL rather than the twoF value.
         detectors : str
@@ -477,7 +481,7 @@ class ComputeFstat(BaseSearchClass):
 
         logging.info('Initialising FstatInput')
         dFreq = 0
-        if self.transient:
+        if self.transientWindowType:
             self.whatToCompute = lalpulsar.FSTATQ_ATOMS_PER_DET
         else:
             self.whatToCompute = lalpulsar.FSTATQ_2F
@@ -593,14 +597,36 @@ class ComputeFstat(BaseSearchClass):
             self.whatToCompute = (self.whatToCompute +
                                   lalpulsar.FSTATQ_2F_PER_DET)
 
-        if self.transient:
+        if self.transientWindowType:
             logging.info('Initialising transient parameters')
             self.windowRange = lalpulsar.transientWindowRange_t()
-            self.windowRange.type = lalpulsar.TRANSIENT_RECTANGULAR
-            self.windowRange.t0Band = 0
-            self.windowRange.dt0 = 1
-            self.windowRange.tauBand = 0
-            self.windowRange.dtau = 1
+            transientWindowTypes = {'none': lalpulsar.TRANSIENT_NONE,
+                                    'rect': lalpulsar.TRANSIENT_RECTANGULAR,
+                                    'exp':  lalpulsar.TRANSIENT_EXPONENTIAL}
+            if self.transientWindowType in transientWindowTypes:
+                self.windowRange.type = transientWindowTypes[self.transientWindowType]
+            else:
+                raise ValueError('Unknown window-type ({}) passed as input. Allowed are: [{}].'.format(self.transientWindowType, ', '.join(transientWindowTypes)))
+
+            self.Tsft = int(1.0/SFTCatalog.data[0].header.deltaF)
+            if self.t0Band is None:
+                self.windowRange.t0Band  = 0
+                self.windowRange.dt0     = 1
+            else:
+                if not isinstance(self.t0Band,int):
+                    logging.warn('Casting non-integer t0Band={} to int...'.format(self.t0Band))
+                    self.t0Band = int(self.t0Band)
+                self.windowRange.t0Band  = self.t0Band
+                self.windowRange.dt0     = self.Tsft
+            if self.tauBand is None:
+                self.windowRange.tauBand = 0
+                self.windowRange.dtau    = 1
+            else:
+                if not isinstance(self.tauBand,int):
+                    logging.warn('Casting non-integer tauBand={} to int...'.format(self.tauBand))
+                    self.tauBand = int(self.tauBand)
+                self.windowRange.tauBand = self.tauBand
+                self.windowRange.dtau    = self.Tsft
 
     def get_fullycoherent_twoF(self, tstart, tend, F0, F1, F2, Alpha, Delta,
                                asini=None, period=None, ecc=None, tp=None,
@@ -624,7 +650,7 @@ class ComputeFstat(BaseSearchClass):
                                self.whatToCompute
                                )
 
-        if self.transient is False:
+        if not self.transientWindowType:
             if self.BSGL is False:
                 return self.FstatResults.twoF[0]
 
@@ -636,7 +662,13 @@ class ComputeFstat(BaseSearchClass):
             return log10_BSGL/np.log10(np.exp(1))
 
         self.windowRange.t0 = int(tstart)  # TYPE UINT4
-        self.windowRange.tau = int(tend - tstart)  # TYPE UINT4
+        if self.windowRange.tauBand == 0:
+            # true single-template search also in transient params:
+            # actual (t0,tau) window was set with tstart, tend before
+            self.windowRange.tau = int(tend - tstart)  # TYPE UINT4
+        else:
+            # grid search: start at minimum tau required for nondegenerate F-stat computation
+            self.windowRange.tau = int(2*self.Tsft)
 
         FS = lalpulsar.ComputeTransientFstatMap(
             self.FstatResults.multiFatoms[0], self.windowRange, False)
@@ -696,8 +728,9 @@ class ComputeFstat(BaseSearchClass):
         max_tau = SFTmaxStartTime - tstart
         taus = np.linspace(min_tau, max_tau, npoints)
         twoFs = []
-        if self.transient is False:
-            self.transient = True
+        if not self.transientWindowType:
+            # still call the transient-Fstat-map function, but using the full range
+            self.transientWindowType = 'none'
             self.init_computefstatistic_single_point()
         for tau in taus:
             twoFs.append(self.get_fullycoherent_twoF(
@@ -868,7 +901,9 @@ class SemiCoherentSearch(ComputeFstat):
 
         self.fs_file_name = "{}/{}_FS.dat".format(self.outdir, self.label)
         self.set_ephemeris_files()
-        self.transient = True
+        self.transientWindowType = 'rect'
+        self.t0Band  = None
+        self.tauBand = None
         self.init_computefstatistic_single_point()
         self.init_semicoherent_parameters()
 
@@ -876,7 +911,7 @@ class SemiCoherentSearch(ComputeFstat):
         logging.info(('Initialising semicoherent parameters from {} to {} in'
                       ' {} segments').format(
             self.minStartTime, self.maxStartTime, self.nsegs))
-        self.transient = True
+        self.transientWindowType = 'rect'
         self.whatToCompute = lalpulsar.FSTATQ_2F+lalpulsar.FSTATQ_ATOMS_PER_DET
         self.tboundaries = np.linspace(self.minStartTime, self.maxStartTime,
                                        self.nsegs+1)
@@ -915,7 +950,7 @@ class SemiCoherentSearch(ComputeFstat):
                                self.whatToCompute
                                )
 
-        #if self.transient is False:
+        #if not self.transientWindowType:
         #    if self.BSGL is False:
         #        return self.FstatResults.twoF[0]
         #    twoF = np.float(self.FstatResults.twoF[0])
@@ -1005,8 +1040,10 @@ class SemiCoherentGlitchSearch(ComputeFstat):
 
         self.fs_file_name = "{}/{}_FS.dat".format(self.outdir, self.label)
         self.set_ephemeris_files()
-        self.transient = True
-        self.binary = False
+        self.transientWindowType = 'rect'
+        self.t0Band  = None
+        self.tauBand = None
+        self.binary  = False
         self.init_computefstatistic_single_point()
 
     def get_semicoherent_nglitch_twoF(self, F0, F1, F2, Alpha, Delta, *args):
diff --git a/pyfstat/grid_based_searches.py b/pyfstat/grid_based_searches.py
index 4a38cbaab11544041e118d9da7ecda07aed11338..daad3b110333bd3b20c585d8cef3336cb3d25154 100644
--- a/pyfstat/grid_based_searches.py
+++ b/pyfstat/grid_based_searches.py
@@ -31,7 +31,8 @@ class GridSearch(BaseSearchClass):
                  Deltas, tref=None, minStartTime=None, maxStartTime=None,
                  nsegs=1, BSGL=False, minCoverFreq=None, maxCoverFreq=None,
                  detectors=None, SSBprec=None, injectSources=None,
-                 input_arrays=False, assumeSqrtSX=None):
+                 input_arrays=False, assumeSqrtSX=None,
+                 transientWindowType=None, t0Band=None, tauBand=None):
         """
         Parameters
         ----------
@@ -48,6 +49,17 @@ class GridSearch(BaseSearchClass):
             GPS seconds of the reference time, start time and end time
         input_arrays: bool
             if true, use the F0s, F1s, etc as is
+        transientWindowType: str
+            If 'rect' or 'exp',
+            compute atoms so that a transient (t0,tau) map can later be computed.
+            ('none' instead of None explicitly calls the transient-window function,
+             but with the full range, for debugging)
+            Currently only supported for nsegs=1.
+        t0Band, tauBand: int
+            if >0, search t0 in (minStartTime,minStartTime+t0Band)
+                   and tau in (2*Tsft,2*Tsft+tauBand).
+            if =0, only compute CW Fstat with t0=minStartTime,
+                   tau=maxStartTime-minStartTime.
 
         For all other parameters, see `pyfstat.ComputeFStat` for details
         """
@@ -66,7 +78,9 @@ class GridSearch(BaseSearchClass):
             self.search = ComputeFstat(
                 tref=self.tref, sftfilepattern=self.sftfilepattern,
                 minCoverFreq=self.minCoverFreq, maxCoverFreq=self.maxCoverFreq,
-                detectors=self.detectors, transient=False,
+                detectors=self.detectors,
+                transientWindowType=self.transientWindowType,
+                t0Band=self.t0Band, tauBand=self.tauBand,
                 minStartTime=self.minStartTime, maxStartTime=self.maxStartTime,
                 BSGL=self.BSGL, SSBprec=self.SSBprec,
                 injectSources=self.injectSources,
@@ -563,7 +577,7 @@ class FrequencySlidingWindow(GridSearch):
         self.search = ComputeFstat(
             tref=self.tref, sftfilepattern=self.sftfilepattern,
             minCoverFreq=self.minCoverFreq, maxCoverFreq=self.maxCoverFreq,
-            detectors=self.detectors, transient=True,
+            detectors=self.detectors, transientWindowType=self.transientWindowType,
             minStartTime=self.minStartTime, maxStartTime=self.maxStartTime,
             BSGL=self.BSGL, SSBprec=self.SSBprec,
             injectSources=self.injectSources)
diff --git a/pyfstat/make_sfts.py b/pyfstat/make_sfts.py
index bf4e258faafca5a30ed042b1da221de3d4a9a963..fda5b60b8d4c8458922ab87d58e7281ac0e0fd4c 100644
--- a/pyfstat/make_sfts.py
+++ b/pyfstat/make_sfts.py
@@ -25,7 +25,8 @@ class Writer(BaseSearchClass):
                  tref=None, F0=30, F1=1e-10, F2=0, Alpha=5e-3,
                  Delta=6e-2, h0=0.1, cosi=0.0, psi=0.0, phi=0, Tsft=1800,
                  outdir=".", sqrtSX=1, Band=4, detectors='H1',
-                 minStartTime=None, maxStartTime=None, add_noise=True):
+                 minStartTime=None, maxStartTime=None, add_noise=True,
+                 transientWindowType='none'):
         """
         Parameters
         ----------
@@ -92,7 +93,7 @@ class Writer(BaseSearchClass):
         self.run_makefakedata()
 
     def get_single_config_line(self, i, Alpha, Delta, h0, cosi, psi, phi, F0,
-                               F1, F2, tref, tstart, duration_days):
+                               F1, F2, tref, window, tstart, duration_days):
         template = (
 """[TS{}]
 Alpha = {:1.18e}
@@ -105,11 +106,11 @@ Freq = {:1.18e}
 f1dot = {:1.18e}
 f2dot = {:1.18e}
 refTime = {:10.6f}
-transientWindowType=rect
-transientStartTime={:10.3f}
-transientTauDays={:1.3f}\n""")
+transientWindowType = {:s}
+transientStartTime = {:10.3f}
+transientTauDays = {:1.3f}\n""")
         return template.format(i, Alpha, Delta, h0, cosi, psi, phi, F0, F1,
-                               F2, tref, tstart, duration_days)
+                               F2, tref, window, tstart, duration_days)
 
     def make_cff(self):
         """
@@ -119,7 +120,7 @@ transientTauDays={:1.3f}\n""")
 
         content = self.get_single_config_line(
             0, self.Alpha, self.Delta, self.h0, self.cosi, self.psi,
-            self.phi, self.F0, self.F1, self.F2, self.tref, self.tstart,
+            self.phi, self.F0, self.F1, self.F2, self.tref, self.transientWindowType, self.tstart,
             self.duration_days)
 
         if self.check_if_cff_file_needs_rewritting(content):
@@ -247,7 +248,8 @@ class GlitchWriter(Writer):
                  delta_F2=0, tref=None, F0=30, F1=1e-10, F2=0, Alpha=5e-3,
                  Delta=6e-2, h0=0.1, cosi=0.0, psi=0.0, phi=0, Tsft=1800,
                  outdir=".", sqrtSX=1, Band=4, detectors='H1',
-                 minStartTime=None, maxStartTime=None, add_noise=True):
+                 minStartTime=None, maxStartTime=None, add_noise=True,
+                 transientWindowType='rect'):
         """
         Parameters
         ----------
@@ -317,7 +319,7 @@ class GlitchWriter(Writer):
                                            self.tbounds[:-1])):
             line = self.get_single_config_line(
                 i, self.Alpha, self.Delta, self.h0, self.cosi, self.psi,
-                t[0], t[1], t[2], t[3], self.tref, ts, d)
+                t[0], t[1], t[2], t[3], self.tref, self.transientWindowType, ts, d)
 
             content += line
 
diff --git a/pyfstat/mcmc_based_searches.py b/pyfstat/mcmc_based_searches.py
index fa13b688b96061e5e3d00f671e777763f127a4b2..d663874c4e80f7c077c6a6929681a89346dc0e20 100644
--- a/pyfstat/mcmc_based_searches.py
+++ b/pyfstat/mcmc_based_searches.py
@@ -76,6 +76,12 @@ class MCMCSearch(core.BaseSearchClass):
         the search
     assumeSqrtSX: float, optional
         Don't estimate noise-floors, but assume (stationary) per-IFO sqrt{SX}
+    transientWindowType: str
+        If 'rect' or 'exp',
+        compute atoms so that a transient (t0,tau) map can later be computed.
+        ('none' instead of None explicitly calls the transient-window function,
+        but with the full range, for debugging)
+        Currently only supported for nsegs=1.
 
     Attributes
     ----------
@@ -108,7 +114,8 @@ class MCMCSearch(core.BaseSearchClass):
                  log10beta_min=-5, theta_initial=None,
                  rhohatmax=1000, binary=False, BSGL=False,
                  SSBprec=None, minCoverFreq=None, maxCoverFreq=None,
-                 injectSources=None, assumeSqrtSX=None):
+                 injectSources=None, assumeSqrtSX=None,
+                 transientWindowType=None):
 
         if os.path.isdir(outdir) is False:
             os.mkdir(outdir)
@@ -150,7 +157,8 @@ class MCMCSearch(core.BaseSearchClass):
         self.search = core.ComputeFstat(
             tref=self.tref, sftfilepattern=self.sftfilepattern,
             minCoverFreq=self.minCoverFreq, maxCoverFreq=self.maxCoverFreq,
-            detectors=self.detectors, BSGL=self.BSGL, transient=False,
+            detectors=self.detectors, BSGL=self.BSGL,
+            transientWindowType=self.transientWindowType,
             minStartTime=self.minStartTime, maxStartTime=self.maxStartTime,
             binary=self.binary, injectSources=self.injectSources,
             assumeSqrtSX=self.assumeSqrtSX, SSBprec=self.SSBprec)
@@ -2195,10 +2203,12 @@ class MCMCTransientSearch(MCMCSearch):
 
     def _initiate_search_object(self):
         logging.info('Setting up search object')
+        if not self.transientWindowType:
+            self.transientWindowType = 'rect'
         self.search = core.ComputeFstat(
             tref=self.tref, sftfilepattern=self.sftfilepattern,
             minCoverFreq=self.minCoverFreq, maxCoverFreq=self.maxCoverFreq,
-            detectors=self.detectors, transient=True,
+            detectors=self.detectors, transientWindowType=self.transientWindowType,
             minStartTime=self.minStartTime, maxStartTime=self.maxStartTime,
             BSGL=self.BSGL, binary=self.binary,
             injectSources=self.injectSources)